; TeX output 2000.07.03:0744Wō>;-ō>=XQ ff cmr12Martin/KreuzerandLorenzoRobbianoVύNH cmbx12ComputationalComm8utativeAlgebraq1*XQ cmr12July3,2000:B궺Springer-V4erlag궽BerlinHeidelbSergNew,YVorkLondonPrarisTVokyoHongWKongBarcelonaBudapSest*Le @궄jbRō>;-ō>;Nff cmbx12FforewordvP+j cmti9Hofstadter'sN cmmi10I2ӲbGeanidealinRaringRboranidealof/aringRDz?Finally*,wedecidedtobGenon-partisanUUandusebGoth.Wō>;VI-ō>;"궲Having.revealedthenamesoftwoofourmainaides,wenowabandonall pretenceiandadmitthattheb}'ook}cisireallyajointe ortofmanypGeople.W*eespGeciallythankAlessioDelPadronewhocarefullycheckedeverydetailofthe=3maintextandtest-solvedalloftheexercises.ThetasksofproGof-readingandXcheckingtutorialswerevqariouslycarriedoutbyJohnAbbGott,AnnaBi-gatti,wMassimoCabGoara,RobertF*orkel,TonyGeramita,BettinaKreuzer,andMarieVitulli.AnnaBigattiwroteorimprovedmanyoftheC0oLCoA"*pro-grams;wepresent,andalsosuggestedthetutorialsabGoutT*oricIdealsandDiophantineH>SystemsandIntegerProgramming.ThetutorialabGoutStrangePolynomials?{comesfromresearchbyJohnAbbGott.ThetutorialaboutElim-ination!ofMoGduleComponentscomesfromresearchinthedoGctoralthesisofMassimoCabGoara.ThetutorialaboutSplineswasconceivedbyJensSchmid-bauer.Mosttutorialsweretested,andinmanycasescorrected,bythestu-dentswhoattendedourlecturecourses.OurcolleaguesBrunoBuchbGerger,Dave*Perkinson,andMossSweedlerhelpGeduswithmaterialforjokesandquotes."Moralhelpcamefromourfamilies.OurwivesBettinaandGabriella,andourchildrenChiara,F*rancesco,Katharina,andVeronikqapatientlyhelpGedustoshouldertheproblemsandburdenswhichwritingabGookentails.AndfromthepracticalpGointofview,thispro8jectcouldneverhavecometoasuccessfulconclusionNwithouttheuntiringsuppGortofDr.MartinPeters,hisassistantRuthUUAllewelt,andtheothermembGersofthesta atSpringerV*erlag."Finally*, wewouldliketomentionourfavouritesoGccerteams,BayernMGunchenjandJuventusT*urin,aswellasthestoGckmarketmaniaofthelate1990s:/theyprovideduswithnever-endingmaterialfordiscussionswhenourworkUUontheb}'ookhbGecameUUtoooverwhelming.^MartinUUKreuzerandLorenzoRobbiano,RegensburgUUandGenovqa,June2000GLe @궄jbRō>;-ō>;궾Contents"V cmbx10F orew9ordp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:feVIn9troQductionҍp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:e1 '0.1>jWhatUUIsThisBoGokAbout??p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.e1'0.2>jWhatUUIsaGrobnerBasis?-͍p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.e2'0.3>jWhoUUInventedThisTheory?p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.e3'0.4>jNow,UUWhatIsThisBoGokR}'eallyUUAbout?ݍp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.e4'0.5>jWhatUUIsThisBoGokNotAbout? Tp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.e7'0.6>jAreUUThereanyApplicationsofThisTheory?wp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.e8'0.7>jHowUUW*asThisBoGokWritten?p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d10'0.8>jWhatUUIsaT*utorial?1op.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d11'0.9>jWhatUUIsC0oLCoA?p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d12'0.10>jAndUUWhatIsThisBoGokGoodfor?:p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d12'0.11>jSomeUUFinalW*ordsofWisdomcp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d131.'F oundationsߊp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p: d15'1.1>jPolynomialUURingsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d17G܅TJutorialN<1.TP9olynomialRepresentationI24p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d24G܅TJutorialN<2.TTheExtendedEuclideanAlgorithmɍp.p.p.p.p.p.p.p.p.p.p.p.p.p. d26G܅TJutorialN<3.TFiniteFieldstp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d27'1.2>jUniqueUUF*actorizationMp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d29G܅TJutorialN<4.TEuclideanDomainszp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d35G܅TJutorialN<5.TSquarefreeP9artsofPolynomials:p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d37G܅TJutorialN<6.TBerlek|ramp'sAlgorithm%p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d38'1.3>jMonomialUUIdealsandMonomialMoGdules{p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d41G܅TJutorialN<7.TCogeneratorsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d47G܅TJutorialN<8.TBasicOpAerationswithMonomialIdealsandModules `48'1.4>jT*ermUUOrderings1{p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d49G܅TJutorialN<9.TMonoidOrderingsRepresen9tedbyMatrices@p.p.p.p.p.p.p.p. d57G܅TJutorialN<10.TClassi cationofT:ermOrderingsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d58'1.5>jLeadingUUT*ermsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d59G܅TJutorialN<11.TP9olynomialRepresentationIAIp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d65G܅TJutorialN<12.TSymmetricP9olynomialsdOp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d66G܅TJutorialN<13.TNewtonP9olytopAes@<p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d67BWō>;VIAII4`Con9tents-ō>;'궲1.6>jTheUUDivisionAlgorithm p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d69 G܅TJutorialN<14.TImplemen9tationoftheDivisionAlgorithmQp.p.p.p.p.p.p.p. d73G܅TJutorialN<15.TNormalRemaindersEp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d75'1.7>jGradings䍍p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d76G܅TJutorialN<16.THomogeneousP9olynomials*p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d832.'Gr@obnerTBasesep:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p: d85'2.1>jSpGecialUUGenerationčp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d87G܅TJutorialN<17.TMinimalP9olynomialsofAlgebraicNumbAersep.p.p.p.p.p. d89'2.2>jRewriteUURulesڍp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d91G܅TJutorialN<18.TAlgebraicNum9bAers:0p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d97'2.3>jSyzygies獍p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p. d99G܅TJutorialN<19.TSyzygiesofElemen9tsofMonomialMoAdulesdrp.p.p.p.p.p.p.c108G܅TJutorialN<20.TLiftingofSyzygies9p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c108'2.4>jGrobnerUUBasesofIdealsandMoGdulesōp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c110>j2.4.A ExistenceUUofGrobnerBases&p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c111>j2.4.B NormalUUF*ormsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c113>j2.4.C ReducedUUGrobnerBases 1p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c115G܅TJutorialN<21.TLinearAlgebra]hp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c119G܅TJutorialN<22.TReducedGr`obnerBasesp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c119'2.5>jBuchbGerger'sUUAlgorithm эp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c121G܅TJutorialN<23.TBuc9hbAerger'sCriterion-6p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c127G܅TJutorialN<24.TComputingSomeGr`obnerBases7p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c129G܅TJutorialN<25.TSomeOptimizationsofBuc9hbAerger'sTAlgorithmVp.p.p.c130'2.6>jHilbGert'sUUNullstellensatzp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c133>j2.6.A TheUUField-TheoreticV*ersion#p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c134>j2.6.B TheUUGeometricV*ersionMp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c137G܅TJutorialN<26.TGraphColouringsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c143G܅TJutorialN<27.TAneV:arietiesp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c1433.'FirstTApplicationsDp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c145'3.1>jComputationUUofSyzygyMoGdules荍p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c148G܅TJutorialN<28.TSplinescp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c155G܅TJutorialN<29.THilbAert'sSyzygyTheorem7'p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c159'3.2>jElementaryUUOpGerationsonModulesp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c160>j3.2.A IntersectionsMލp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c162>j3.2.B ColonUUIdealsandAnnihilators܍p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c166>j3.2.C ColonUUMoGdules p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c169G܅TJutorialN<30.TComputationofIn9tersectionsލp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c174G܅TJutorialN<31.TComputationofColonIdealsandColonMoAdulesp.p.c175'3.3>jHomomorphismsUUofMoGdulesp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c177>j3.3.A Kernels,UUImages,andLiftingsofLinearMaps荍p.p.p.p.p.p.p.p.c178>j3.3.B Hom-MoGdulesp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c181G܅TJutorialN<32.TComputingKernelsandPullbac9ksmp.p.p.p.p.p.p.p.p.p.p.p.p.p.c191G܅TJutorialN<33.TTheDepthofaMoAdule3p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c192 BʠWō>;Con9tentsJIX-ō>;'궲3.4>jElimination?p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c195 G܅TJutorialN<34.TEliminationofMoAduleComponen9tsp.p.p.p.p.p.p.p.p.p.p.p.p.c202G܅TJutorialN<35.TProjectiv9eSpacesandGramanniansC6p.p.p.p.p.p.p.p.p.p.p.p.c204G܅TJutorialN<36.TDiophan9tineSystemsandIntegerProgramming ip.p.p.c207'3.5>jLoGcalizationUUandSaturationcp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c211>j3.5.A LoGcalization?p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c212>j3.5.B Saturationp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c215G܅TJutorialN<37.TComputationofSaturationsqp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c220G܅TJutorialN<38.TT:oricIdeals؍p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c221'3.6>jHomomorphismsUUofAlgebrasqOp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c225G܅TJutorialN<39.TProjections=;p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c234G܅TJutorialN<40.TGr`obnerBasesandIn9v|rariantTTheoryYp.p.p.p.p.p.p.p.p.p.p.p.c236G܅TJutorialN<41.TSubalgebrasofF:unctionFields,p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c239'3.7>jSystemsUUofPolynomialEquations8[p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c241>j3.7.A AUUBoundfortheNumbGerofSolutionsp.p.p.p.p.p.p.p.p.p.p.p.p.p.c243>j3.7.B RadicalsUUofZero-DimensionalIdeals\p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c246>j3.7.C SolvingUUSystemsE ectivelyp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c254G܅TJutorialN<42.TStrangeP9olynomialslp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c261G܅TJutorialN<43.TPrimaryDecompAositionsp.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c263G܅TJutorialN<44.TMoAdernP9ortfolioTheory;p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.p.c267A.'Ho9wTtoGetStartedwithCEo1CoFlAp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c275B.'Ho9wTtoProgramCEo1CoFlA?p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c283C.'ATP9otpQourriofCEo1CoFlATPrograms荑p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c293D.'Hin9tsTforSelectedExercisesp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c305Notationap:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c309Bibliograph9y~)p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c313IndexMp:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:p:c315xLe @궄jbRō>;-ō>;궾Intros3ductionvsIt#seemstobeacommonpracticeofbookreadersto ئglance1throughtheintroductionandskiptherest.TJoN;24`In9troAduction-ō>;"궲AlthoughܸthefundamentalideasofComputationalCommutativeAlgebra are):deeplyroGotedinthedevelopment):ofmathematicsinthe):20^ٓRcmr7th 3century*,their-fullpGower-onlyemergedinthelasttwenty-years.OnecentralnotionwhichşembGodiesboththeoldandthenewfeaturesofthissub8jectisthenotionUUofaGrobnerb}'asis.30.2**WhatIsaGr@obnerBasis?궲TheitheoryofGrobnerbasesisawonderfulexampleofhowanideausedtosolve8~oneproblemcanbGecomethekeyforsolvingagreatvqarietyofotherproblems3indi erentareasofmathematicsandevenoutsidemathematics.TheintroGductionofGrobnerbasesisanalogoustotheintroGductionofѵiasa8solutionoftheequation8x^2蕲+l"1G<=0.8AfterihasbGeenaddedtothereals,the_T eldofcomplexnumbGers_Tarises.TheastonishingfactisthatinthiswaynotUonlyUx^2T+1=0hasUasolution,butalsoeveryotherpGolynomialequationoverUUtherealshasasolution."SuppGoseUUnowthatwewanttoaddressthefollowingproblem.Let\f1|s(x1;:::;x 0ercmmi7nq~)=0;:::;fsF:(x1|s;:::;xnq~)=0bGe3asystemofpolynomialequationsde nedover3anarbitrary eld,andlet 궵f(x1|s;:::;xnq~)=0PbGePanadditionalpolynomialequation.Howcanwedecideif!f(x1|s;:::;xnq~)=0holds!forallsolutionsoftheinitialsystemofequations?Naturally*,thisdepGendsonwherewelookforsuchsolutions.Inanyevent,partoftheproblemiscertainlytodecidewhetherfbGelongstotheidealI궲generatedNbyNf1|s;:::;fsF:,i.e.whethertherearepGolynomialsg1|s;:::;gs esuchthatfڧ=g1|sf1 +: !", cmsy10+:gsF:fsF:.Iff2I,theneverysolutionoff1C=8ײ=fs R=0isUUalsoasolutionofUUfڧ=0."TheproblemofdecidingwhetherornotfmѸ2ZBIviscalledtheIde}'al#Mem-b}'ershipxProblem.wItcanbGeviewedasthesearchforasolutionofwx^23+O1H=0in57ouranalogy*.AsinthecaseoftheintroGductionof57i,oncethekeytoGol,namely۠aGrobnerbasisof۠I,hasbGeenfound,wecansolvenotonlytheIdealMembGershipUUProblem,butalsoavqastarrayofotherproblems."Now,NPwhatisaGrobnerbasis?ItisaspGecialsystemofgeneratorsoftheidealI]withthepropGertythatthedecisionastowhetherornotfD%20I]canbGe9answeredbyasimpledivisionwithremainderproGcess.ItsimportanceforpracticalK+computercalculationscomesfromthefactthatthereisanexplicitalgorithm,calledBuchbGerger'sAlgorithm,whichallowsusto ndaGrobnerbasisUUstartingfromanysystemofgeneratorsUUff1|s;:::;fsF:gofI.#Wō>;70.3WhoTIn9ventedThisTheory?y3-ō>;0.3**WhoInventedThisTheory?궲AsY oftenhappGens,therearemanypeoplewhomaylayclaimtoinventing somedaspGectsofthistheory*.Inourview,thema8jorstepwastakenbyB.BuchbGerger°inthemid-sixties.HeformulatedtheconceptofGrobnerbasesand,extendingasuggestionofhisadvisorW.Grobner,foundanalgorithmtocompute`them,andproved`thefundamentaltheoremonwhichthecorrectnessandUUterminationofthealgorithmhinges."F*orGmanyyearstheimpGortanceofBuchbGerger'sworkwasnotfullyap-preciated.ZOnlyintheeightiesdidresearchersinmathematicsandcomputerscienceWstartadeepinvestigationWofthenewtheory*.ManygeneralizationsandawidevqarietyofapplicationsweredevelopGed.IthasnowbGecomeclearthatthetheoryofGrobnerbasescanbGewidelyusedinmanyareasofscience.TheLsimplicityofitsfundamentalideasstandsinstarkcontrasttoitspGowerandYthebreadthofitsapplications.SimplicityandpGower:twoingredientswhichUUcombinepGerfectlytoensurethecontinuedsuccessofthistheory*."F*or.xinstance,researchersincommutativealgebraandalgebraicgeometrybGene ttedwimmediatelyfromtheappearanceofspecializedcomputeralgebrasystems͑suchasC0oLCoA,Macaulay*,andSingular.Basedonadvqancedimple-mentationsfofBuchbGerger'sAlgorithmforthecomputationofGrobnerbases,theyallowtheusertostudyexamples,calculateinvqariants,andexploreob-jectsM3onecouldonlydreamofdealingwithbGefore.Themostfascinatingfea-tureofthesesystemsisthattheircapabilitiescomefromtyingtogetherdeepideasUUinbGothmathematicsandcomputerscience."It*wasonlyintheninetiesthattheproGcessofestablishingcomputeral-gebraqOasanindepGendentdisciplinestartedtotakeplace.Thiscontributedagreat~dealtotheincreaseddemandtolearnabGoutGrobnerbasesandinspiredmany+}authorstowritebGooks+}aboutthesub8ject.F*orinstance,amongothers,theUUfollowingbGooksUUhavealreadyappGeared.1)%W.UUAdamsandP*.Loustaunau,AnIntr}'oductiontoGrobnerBases2)%T.UUBeckerandV.W*eispfenning,GrobnerBases3)%B.UUBuchbGergerandF.Winkler(eds.),GrobnerBasesandApplic}'ations4)%D.UUCox,J.LittleandD.O'Shea,Ide}'als,V;arietiesandAlgorithms5)%D.'Eisenbud,CommutativeFuAlgebr}'awithaViewtowardAlgebraicGeom-%etry,UUChapter156)%B.UUMishra,Algorithmicalgebr}'a7)%W.V*asconcelos,ComputationalYMetho}'dsinCommutativeAlgebraand%Algebr}'aicGeometry8)%F.UUWinkler,PolynomialAlgorithmsinComputerAlgebr}'a9)%R.UUF*robGerg,AnIntr}'oductiontoGrobnerBases"궲IsthereanyneedforanotherbGookonthesub8ject?Clearlywethinkso.F*orF+theremainderofthisintroGduction,weshalltrytoexplainwhy*.FirstweshouldexplainhowthecontentsofthisbGookrelatetothebGookslistedabGove.(Wō>;44`In9troAduction-ō>;0.4**Now,WhatIsThisBo`ok!F C cmbxti10RKeallyAbout?궲InsteadofdwellingongeneralitiesandthevirtuesofthetheoryofGrobner bases,letusgetdowntosomenitty-grittydetailsofr}'ealcmathematics.Letusexaminesomeconcreteproblemswhosesolutionsweshalltrytoexplainin=thisbGook.=F*orinstance,letusstartwiththeIdealMembGershipProblemmentionedUUabGove."SuppGoseZwearegivenapGolynomialringZPޤ={K[x1|s;:::;xnq~]overZsome eld.K,.apGolynomialfe2ֵPc,andsomeotherpGolynomialsf1|s;:::;fsM2ֵP궲whichUUgenerateanidealUUI=(f1|s;:::;fsF:)Pc.ڪQuestionT1O#Howc}'anwedecidewhetherfڧ2I?"궲Inotherwords,weareaskingwhetheritispGossibleto ndpolynomials 궵g1|s;:::;gs R2P2suchthatfڧ=g1f1c+~ y+~gsF:fsF:.Insucharelation,manytermscancancelontheright-handside.ThusthereisnoobviousaprioribGoundonKthedegreesofKg1|s;:::;gsF:,andwecannotsimplyconvertthisquestiontoaUUsystemoflinearequationsbycomparingcoGecients.Ǎ"NextwesuppGosewearegivena nitelygeneratedK-algebraRospGeci edby{ generatorsandrelations.Thismeansthatwehavearepresentation{ R=궵PV=I7withUUPandIasUUabGove.ڪQuestionT2O#Howc}'anweperformadditionandmultiplicationinR?"궲Ofcourse,iff1|s;f2ܸ2"iP﮲arerepresentativesofresidueclassesr1|s;r2ܸ2"iRDz, then0f1+rf2 M(resp.f1|sf2|s)0representstheresidueclass0r1+rr2 M(resp.r1|sr2|s).ButthisdepGendsonthechoiceofrepresentatives,andifwewanttocheckwhethertwodi erentresultsrepresentthesameresidueclass,weareledbackto Question1.A much bGettersolutionwouldbetohave a\canonical"repre-sentative1foreachresidueclass,andtocomputethecanonicalrepresentativeofUUr1S+8r2Ȳ(resp.r1|sr2|s)."Moregenerally*,wecanaskthesamequestionformoGdules.IfMisa nitelyoOgeneratedoORDz-moGdule,thenMjisalsoa nitelygeneratedPc-moGdulevia)thesurjectivehomomorphism)P ( msam10)#RDz,and,usinggeneratorsandrela-tions,themoGduleMhasapresentationoftheformMT͍Ƹ+3Ʋ=oPc^r1=qNforsome궵Pc-submoGduleUUN3Pc^r1.ڪQuestionT3O#HowGc}'anweperformadditionandscalarmultiplicationinGM?"궲Let)usnowturntoadi erentproblem.F*orpGolynomialsinoneindetermi- nate, thereisawell-known andelementaryalgorithmfordoingdivisionwithremainder.IfwetrytogeneralizethistopGolynomialsinnindeterminates,weUUencounteranumbGerofdiculties.ڪQuestionT4O#HowBc}'anweperformpolynomialdivisionforpolynomialsinBnindeterminates?SInotherwor}'ds,istherea\canonical"representationSеf6P=궵q1|sf1+K=+KqsF:fs΅+psuchthatq1;:::;qs޸2Peandther}'emainderp2Peis\small"?DWō>;+0.4No9w,TWhatIsThisBoAokRealxlyTAbout?y5-ō>;"궲Again%Ywe ndaconnectionwithQuestion2.Ifwecande nethepGolyno- mialZdivisioninacanonicalway*,ZwecantrytousetheremainderZpasthecanonicalrepresentativeoftheresidueclassoffinRDz.EvenifweareabletopGerformqthebasicoperationsinqRdorM,qthenextstephastobethepossi-bility@ofcomputingwithideals(resp.submoGdules).SupposewehavefurtherpGolynomialsUUg1|s;:::;gtLn2PwhichUUgenerateanidealJQ=(g1|s;:::;gtV).QuestionT5O#Howc}'anweperformelementaryoperationsonidealsorsub-mo}'dules?&Moreprecisely,howcanwecomputesystemsofgeneratorsofthefollowingide}'als?~oa)&I¸\8JHb)&I:lO \cmmi5P ƵJQ=ffڧ2P*jfLo8JIgHc)&I:lP ƵJ9^ O!cmsy71 66=ffڧ2P*jfLo8J9^iI7forUUsome,zi2 msbm10Ng"궲ThezcasesofcomputingzI+QJpɲandIQJpɲarezobviouslyeasy*.ItturnsoutthatthekeystothesolutionofthislastquestionaretheanswerstoournextBtwoproblems,namelytheproblemsofcomputingsyzygymoGdulesandeliminationUUmoGdules.QuestionT6O#How<c}'anwecomputethemoduleofallDsyzygiesof<(f1|s;:::;fsF:),i.e.thePc-mo}'dule13Syz@RqƴPGA(f1|s;:::;fsF:)=f(g1;:::;gsF:)2Pcspjg1f1S+8g+8gsF:fs R=0g?QuestionT7O#Howc}'anwesolvetheEliminationProblem,i.e.for1m;64`In9troAduction-ō>;"궲NowJsuppGoseJҵR߲=PV=IandSZ=Q=JA areJtwo nitelygeneratedJҵK-alge- bras,rwhererQ=K[y1|s;:::;ym]isranotherpGolynomialringandJ<Qisanideal.F*urthermore,suppGosethatŵ ":R߸U!S3RisaK-algebrahomomorphismwhichtisexplicitlygivenbyalistofpGolynomialsintQrepresentingttheimages궵 [ٲ(x1S+8I);:::; (xn^+I).ȍQuestionT10THowģc}'anwecomputepresentationsofthekernelandtheimageof [? Andhowc}'anwedecideforagivenelementof SXwhetheritisintheimageof [?"궲Finally*,oneofthemostfamousapplicationsofComputationalCommu-tativeUUAlgebraisthepGossibilitytosolvepGolynomialsystemsofequations.QuestionT11THow,c}'anwecheckwhetherthesystemofpolynomialequationsaevf1|s(x1;:::;xnq~)=8ײ=fsF:(x1|s;:::;xnq~)=0has2solutionsin2}fe 5UKn6,wher}'e2}fe 5UKwisthealgebraicclosureof2K,andwhether thenumb}'erofthosesolutionsis niteorin nite?QuestionT12TIfthesystemofp}'olynomialequationsaevf1|s(x1;:::;xnq~)=8ײ=fsF:(x1|s;:::;xnq~)=0hasonly nitelymanysolutions(a1|s;:::;anq~)[2}fe 5UKQn,howc}'anwedescribe them?F;orinstanc}'e,canwecomputetheminimalpolynomialsoftheelements궵a1|s;:::;an )overNK?NAndhowc}'anwetellwhichofthecombinationsofthezer}'osofthosepolynomialssolvethesystemofequations?"궲TheseandmanyrelatedquestionswillbGeansweredinthisbGook.F*orasimilar9descriptionofthecontents9ofV*olume2wereferthereadertoitsintroGduction.mnHereweonlymentionthatitwillcontainthreemorechapterscalledsChapterUUIVU[TheUUHomogeneousCasesChapterUUVU[HilbGertUUF*unctionssChapterUUVIU[F*urtherUUApplicationsofGrobnerBases"LetusendthisdiscussionbypGointingoutoneimpGortantchoicewemade.F*romXtheverybGeginningwehavedevelopGedthetheoryforsubmodulesoffree)moGdulesover)polynomialrings,andnotjustfortheirideals.Thisdi ersmarkedly?{fromthecommonpracticeofintroGducingeverythingonlyinthecase#ofidealsandthenleavingtheappropriategeneralizationstothereader."Naturally*,'Fthereisatrade-o involved'Fhere.Wehave'Ftopayforourgenerality>withslightcomplicationslingeringaroundalmosteverycorner.Thissuggeststhattheusualexercises\lefttothereader"byotherauthorscouldharbGourafewnastymines.ButmuchmoreimpGortantly*,inourviewGrobnerGbasistheoryisintrinsic}'allygIJabGoutmodules.Buchberger'sGAlgorithm,hisGrobnerbasiscriterion,andothercentralnotionsandresultsdealwith}Wō>;L0.5WhatTIsThisBoAokNotAbout?y7-ō>;syzygies.!kInanycase,thesetofallsyzygiesisamoGdule,notanideal.There- fore&apropGerintroductiontoGrobnerbasistheorycannotavoid&submodulesoffreemoGdules.Infact,webelievethisbookshowsthatthereisnoreasontoUUavoidthem."Finally*,4uwewouldliketopGointoutthatevenifyouareonlyinterestedinthetheoryofpGolynomialideals,oftenyoustillhavetobGeabletocomputewithtmoGdules,forinstanceifyouwanttocomputesomeinvqariantswhicharederivedfromthefreeresolutionoftheideal.WithoutmoGdules,anumbGerof4impGortantapplicationsofthistheorywouldhavetoremainconspicuouslyabsent!-60.5**WhatIsThisBo`okNotAbout?궲ThelistoftopicswhichwedonotutalkabGoutistoolongtobeincludedhere,butMforinstanceitcontainssoGccer,chess,gardening,andourotherfavouritepastimes."Computationalv=CommutativeAlgebraispartofalarger eldofinves-tigation Dcalledsymb}'olic9qcomputationͧwhich DsomepGeoplealsocallc}'omputeralgebr}'a.lCoveringthishugetopicisbGeyondthescopGeofourbook.So,whatisrsymbGoliccomputationabout?Abstractlyspeaking,itdealswiththoseal-gorithmsX>whichallowmoGderncomputerstoperformcomputationsinvolvingsymbGols,UUandnotonlynumbGers."Unlikenkyourmathteacher,computersdonotob8jecttosymbGolicsimpli -cationUUandrewritingofformulassuchasፍ&hS6n4SʉfeΟ16n)`=4and&h9O9n59OʉfeΟ19nE[Ų=5and 9O19O&fes6@2+8bu cmex10 j1j&fes3i l1l&fes2b(=b 1&fes6+ l1l&fes3bi8b j1j&fes6+ l1l&fes2b궲MoreBseriously*,symbGoliccomputationincludestopicssuchascomputationalgroupYtheory*,symbGolicintegration,symbGolicsummation,quanti erelimina-tion,UUetc.,whichweshallnottouchhere."Anotherkcircleofquestionswhichweavoidisconcernedwithcomputabil-ity*,Urecursivefunctions,decidability*,andsoon.AlmostallofouralgorithmswillQbGeformulatedforpolynomialsandvectorsofpolynomialswithcoe-cientsFinanarbitrary eld.Clearly*,ifyouwanttoimplementthosealgo-rithms!onacomputer,youwillhavetoassumethatthe eldisc}'omputable.ThisMmeans(approximately)thatyouhavetobGeabletostoreanelementofyour< eldin nitelymanymemorycellsofthecomputer,thatyoucancheckin> nitelymanystepswhethertwosuchrepresentationscorrespGondtothesame eldelement,andyouhavetoprovidealgorithmsforpGerformingthefourHbasicopGerationsH+;;;,Hi.e.sequencesofinstructionswhichperformtheseUUopGerationsin nitelymanysteps."F*orxvus,thisassumptiondoGesnotpresentanyproblematall,sinceforconcrete)implementationsweshallalwaysassumethatthebase eldisoneofUUthe eldsimplementedinC0oLCoAzL,andthose eldsarecomputable.Wō>;84`In9troAduction-ō>;"궲Moreover,(wearenotgoingtogiveadetailedaccountofthehistoryofthe topicsjLwediscuss.Likewise,althoughattheendofthebGookjLyouwill ndsomereferences,QDwedecidednottociteeverythingeverywhere.Morecorrectly*,wedidVnotciteanythinganywhere.IfyouwantadditionalinformationabGouttheThistoricaldevelopment,TyoucanloGokintothebGooksmentionedabove.F*orspGeci creferencestorecentresearchpapGers,werecommendthatyouuseelectronicwpreprintandreviewservices.ThenumbGerofpapersinComputa-tional\>CommutativeAlgebraisgrowingexpGonentially*,andunlikeGrobnerbasisncomputations,itdoGesnotseemlikelythatitwillendeventually*.IfallelseUUfails,youcanalsodropane-mailtous,andwewilltrytohelpyou."Finally*,JyouareinterestedinpracticalapplicationsofComputationalCom-mutativeoAlgebra,thecomplexityissuesyouaregoingtoencounterareofadi erent2natureanyway*.Usually,theycannotbGesolvedbytheoreticalconsid-erations.eInstead,theyrequireagoGodegraspoftheunderlyingmathematicalproblemUUandaconcertede orttoimproveUUyourprogramcoGde.2t0.6**AreThereanyApplicationsofThisTheory?U궲De nitely*,ݼyes!ComputationalCommutativeAlgebrahasmanyapplications,some&oftheminotherareasofmathematics,andsomeoftheminothersciences.٦Amongstothers,weshallseesomeeasycasesofthefollowingap-plications.պApplicationsTinAlgebraicGeometryݍ%HilbGert'sUUNullstellensatz(seeSection2.6)%AneUUvqarieties(seeT*utorial27) Wō>;{Z0.6AreTTherean9yApplicationsofThisTheory?y9-ō>;%Pro8jectiveUUspacesandGramannians(seeT*utorial35) %Saturationc(forcomputingthehomogeneousvqanishingidealofapro8jec-%tiveUUvqariety*,seeSection3.5andVolume2)%SystemsUUofpGolynomialequations(seeSection3.7)%PrimarydecompGositions(forcomputingirreduciblecomponentsofvqari-%eties,UUseeT*utorial43)%Pro8jectiveUUV*arieties(seeVolume2)%HomogenizationUU(forcomputingpro8jectiveclosures,seeV*olume2)%Set-theoreticUUcompleteintersections(seeV*olume2)%DimensionsUUofaneandpro8jectivevqarieties(seeV*olume2)%IdealsUUofpGoints(seeV*olume2)ApplicationsTinNum9bQerTheory%MoGdularaarithmetic,factoringpolynomialsovera nite elds(seeT*utori-%alsUU3and6)%Computationsinthe eldofalgebraicnumbGers(seeT*utorials17and18)%MagicUUsquares(seeV*olume2)ApplicationsTinHomologicalAlgebra%ComputationUUofsyzygymoGdules(seeSection3.1)%Kernels,]imagesandliftingsofmoGdulehomomorphisms(seeSection3.3)%ComputationUUofHom-moGdules(seeSection3.3)%Ext-moGdulesUUandthedepthofamodule(seeT*utorial33)%GradedUUfreeresolutions(seeV*olume2)ApplicationsTinCom9binatorics%MonomialUUidealsandmoGdules(seeSection1.3)%GraphUUcolourings(seeT*utorial26)%T*oricUUideals(seeTutorial38)PracticalTandOtherApplications%SplinesUU(seeT*utorial28)%DiophantineSystemsandIntegerProgramming(seeT*utorial36and38)%StrangeUUPolynomials(seeT*utorial42)%MathematicalUUFinance:MoGdernPortfolioTheory(seeT*utorial44)%PhotogrammetryUU(seeV*olume2)%ChessUUPuzzles(seeV*olume2)%Statistics:UUDesignofExpGeriments(seeV*olume2)%AutomaticUUTheoremProving(seeV*olume2) Wō>;104`In9troAduction-ō>;0.7**HowWasThisBo`okWritten?z궲In6ouropinion,anyplanforwritingabGook6shouldincludeasetofruleswhich the~authorsintendtofollowconsistently*.Thismetaruleismorediculttocomplywiththanonethinks,andindeedmanybGooksappeartohavebGeenwritteninamorelibGeralmanner.StrictlyfollowingasetofrulesseemstobGeJincontrastwiththefreedomofchoGosingdi erentapproachestodi erentproblems.XOntheotherhand,toGomuchXfreedomsometimesleadstosituationswhich,UUinouropinion,che}'atthereader."F*or vinstance,oneofourmostimpGortantrulesisthatstatementscalledLemma,PropGosition,Theorem,etc.havetobefollowedbyacompleteproGof,andthedevelopmentofthetheoryshouldbGeasself-containedaspossible.In5particular,weavoidrelegatingproGofstoexercises,givingaproofwhichconsistsofareferencewhichisnotspGeci c,givingaproofwhichconsistsof[areferencehardtoverify*,bGecauseitusesdi erentassumptionsand/ornotation,andgivingaproGofwhichconsistsofareferencetoalaterpartoftheUUbGook."AnotherfundamentalruleisthatthenotationusedinthisbGookiscon-sistentthroughoutthebGookandalwaysascloseaspGossibletothenotationofthecomputeralgebrasystemC0oLCoA.Itisclearthat,inanemerging eldlikecomputeralgebra,thenotationisstillin uxandfewconventionsholduniformly*.xWethinkthatthesituationincomputeralgebraisevenworsethantelsewhere.JustloGokatthefollowingtablewhichpresentsthedi erentterminologies^andthenotationusedforsomefundamentalob8jectsinourref-erences"rlistedinSubsection0.3.ItssecondrowcontainsourchoiceswhichagreeUUwithC0oLCoAzL."GivenՐanon-zeropGolynomialՐfinapolynomialringՐK[x1|s;:::;xnq~]andanorderingβonthesetofproGductsofpowersofindeterminates,welet궵x: 1l1 Dx^ n፴nbGe 1thelargestelement(withrespectto 1[ٲ)inthesupportof 1f궲andUUc2K qitsUUcoGecientinf.U궟ffK1fd͟ ff}+ ff4}x: 1l1 Dx^ n፴n* ffxNotation џ ffZPc8x: 1l1 D:::x^ n፴n ffQNotation ] ffffK1ffK1͟ ff}+ ff0JleadingUUterm3 ffL*TD(f)  ffG?(none)> ffYLM,:2&в(f)  ffffK1ffK1ͤ ffΟfd1)͡ ffleadingUUpGowerproGduct͟ fflp(f)` ffZleadingUUterm崟 ff$jlt+(f))A ffffK1ͤ ffΟfd2)͡ ff5headUUterm {  ffHTC(f) ffheadUUmonomial  ffjHM0(f) )A ffffK1ͤ ffΟfd3)͡ ffleadingUUpGowerproGduct͟ ffLPPuk5l(f)П ffsleadingUUmonomial͟ ffLM,2Յ(f) i@ ffffK1ͤ ffΟfd4)͡ ff%1leadingUUmonomial ff1LM(f)E ffZleadingUUterm崟 ff!ngL*T.(f),П ffffK1ͤ ffΟfd5)͡ ff'initialUUmonomial" ff(none)` ffεvinitialUUtermП ff 5in(ٟ>/Jڲ(f)  ffffK1ͤ ffΟfd6)͡ ff5headUUterm {  ff܅Hterm(f)_ ffheadUUmonomial  ffdHmono7q(f)͟ ffffK1ͤ ffΟfd7)͡ ff'initialUUmonomial" ffKinc(f);M(f)"% ffZleadingUUterm崟 ff1ltܟ(f);L(f)\ ffffK1ͤ ffΟfd8)͡ ffleadingUUpGowerproGduct͟ ffilpp(f) C ffڪinitial' ff#in+(f)S ffffK1ͤ ffΟfd9)͡ ff%1leadingUUmonomial ffXlmuk(f)lџ ffZleadingUUterm崟 ff$jlt+(f))A ffffK1"Afurtherconstraintisthatwehavetriedtostructureeachsectionac- cordingUUtothefollowingscheme:introGduction,body*,exercises,tutorials. Wō>;%0.8WhatTIsaT:utorial?ٮ11-ō>;"궲TheintroGductiondescribesthecontentinalivelystyle,whereItalian 궲imagination overtakesGermanrigour.Metaphors,sketchesofexamples,andpsychologicalmotivqationsofthethemesofthesectionareincludedhere.ThebGody,#isthetechnicalpartofthesection.Itincludesde nitions,theorems,proGofs,etc.V*eryfewcompromiseswithimaginationareacceptedhere.How-ever,UUwealwaystrytolivenupthetextbyincludingexamples."Nothing~=spGecialneedstobesaidabouttheexercises,exceptmaybethatthey9=aresupp}'osedAƲto9=bGeeasy*.A96carefulreaderofthebookshouldsucceedinsolvingthem,andtomakelifeeveneasier,weincludesomehintsforselectedexercisesinthetext,andsomemoreinAppGendixD.ThenthereisoneofthemainnfeaturesofthisbGooknwhichwebGelievetobGenon-standard.AttheendofUUeverysectiontherearetutorials.30.8**WhatIsaTutorial?궲Almost2allbGooks2aboutcomputeralgebraincludesomeexerciseswhichre-quire6{thatactualcomputationsbGeperformedwiththehelpofacomputeralgebralsystem.Butinouropinion,thegapbGetweenlthetheoryandactualcomputationsUUismuchUUtoGowide."Firstofall,thealgorithmsinthetextareusuallypresentedinpseudo}'code궲which,ingeneral,iscompletelydi erentfromthewayyouwriteafunctioninacomputeralgebrasystem.Infact,wehaveahardtimeunderstandingprecisely~whatpseudoGcode~is,bGecauseitisnotrigorouslyde ned.Instead,wehaveVtriedtopresentallalgorithmsinthesamewaymathematiciansformu-lateqYothertheoremsandtoprovideexplicitandcompleteproGofsoftheir nite-ness#andcorrectness.Ifthereaderisaskedtoimplementacertainalgorithmashapartofsometutorialorexercise,thesenaturallanguagedescriptionsshouldUUtranslateeasilyintocomputercoGdeonastep-by-stepbasis."Secondly*,NtonarrowthegapbGetweentheoryandcomputationevenmore,we\decidedtolinkthetutorialsandsomeexerciseswithaspGeci ccomputeralgebra$system,namelyC0oLCoA .ThisdoGesnotmeanthatyoucannotuseanotherUcomputeralgebrasystem.Itonlymeansthattherede nitelyisasolutionUUusingC0oLCoAzL."EveryCtutorialdevelopsatheme.Sometimesweanticipatelaterpartsoftheutheory*,orwestepoutalittlefromthemainstreamandprovidesomepGointers˦toapplicationsorotherareasofinterest.AˇtutorialislikeasmallsectionobyitselfwhichisnotusedinthemaintextofthebGook.oSomee ortonEXthepartofthereadermaybGerequiredtodevelopasmallpieceoftheoryor*toimplementcertainalgorithms.However,manysuggestionsandhintsintheUUC0oLCoA!ϡstyleUUaretheretoguideyouthroughthemaindiculties. 1٠Wō>;124`In9troAduction-ō>;0.9**WhatIsC>oCoA?þ궲C0oLCoA2bisS0acomputeralgebrasystem.ItisfreelyavqailableandmaybGefound onUUtheinternetattheURLv";R0.11SomeTFinalW:ordsofWisdomٮ13-ō>;궲proGofs, andgeneroushintsforexercisesandtutorialswhichshouldhelpto paveC yourroad.ThisdoGesnotnecessarilymeanthatwhenyouworkyourwayUUthroughthebGook,UUtherewillbGenounexpecteddiculties."Probably youalreadyknowsomeofthetopicswediscuss.Or,maybGe,youthinkHyouknowthem.F*orinstance,youmayhavepreviouslyencounteredthepGolynomialringinasingleindeterminateovera eldsuchasQ,R,oreC,eandyoumayfeelcomfortableusingsuchpGolynomials.ButdidyouknowthattherearepGolynomialswhosesquarehasfewertermsthanthepGolynomialitself?At rstglancethisseemsunlikely*,atsecondglanceitmayloGokmpossible,andatthirdglanceyouwillstillnotbeabletodecide,becauseyouUU ndnoexample.ByloGokingatthepolynomial&^Gf/W-=:Kx12 Ʋ+ l2l&fes5 Fax11 jM2l&fe25BԵx10+ h4l&fe Y125?Gx9S h2l&fe Y125x8+ h2l&fe Y125x7:K -333&fe̟2750ڵx6S h1l&fe Y275?Gx5+ f1l&fe̟1375;x4 f2l&fe̟6875x3+ f1l&fe̟6875x2 f1l&fe̟6875x8 d1l&fe?13750궲whoseUUsquareis궵f2 2=0x24 Ʋ+ l4l&fes5 Fax23+ h44l&fe̟3125;x19+ h2441l&fe겟1718754x18 h2016l&fe겟171875x17 f16719l&fe㘟37812500&-x12#K+ ,14133&fe%94531253x11 Ƹ c33l&fe겟8593754x7S+ c313l&fe%8593750"1x6+ al1l&fe%4296875x5+ _1l&fe㘟47265625&-x8+ ]߱1l&fe# 189062500궲youcanconvinceyourselfthatsuchaphenomenonactuallyoGccurs.Butwhat isreallysurprisingisthatthisisthesimplestbexamplepGossible,asweshallseeUUinT*utorial42."ThusU|weadviseyoutogothroughthebGookwithanopenandcriticalmind.;W*ehave;triedto llitwithalotofhiddentreasures,andwethinkthateven }ifyouhavesomepreviousknowledgeofComputationalCommutativeAlgebra,youwill ndsomethingneworsomethingthatcouldchangeyourviewUUofonetopicoranother."Last,butnotleast,thebGookcanalsobGeusedasarepositoryofexplicitalgorithms,}programmingexercises,andC0oLCoA" Ytricks.So,evenifcomputersandprogrammingenticeyoumorethanalgebraictheorems,youwill ndplentyUUofthingstolearnandtodo.30.110SomeFinalWordsofWisdom궲Naturally*,thisintroGductionhastoleavemanyimpGortantquestionsunan-swered.WhatisthedeepGermeaningofComputationalCommutativeAl-gebra?WhatistherelationshipbGetweendoingcomputationsandprovingalgebraic]theorems?Willthistheory ndwidespreadapplications?WhatisthefutureofComputationalCommutativeAlgebra?InsteadofelabGoratingontheseprofoundphilosophicalproblems,letusendthisintroGductionandsendUUyouo intoChapter1withafewwordsofwisdombyMarkGreen.MsWō>;144`In9troAduction-ō>;%Thereisonechangewhichhasovertakencommutativealgebrathatisin %myP*viewrevolutionaryincharacter{theadventofsymboliccomputation.%This?isasyetanun nishedrevolution.AÎtpresent,manyresearchersrou-%tinely6useMacaulay,Maple,Mathematica,andC'oZ:CoA]toperformcomputer%experiments,andasmorepeoplebecomeadeptatdoingthis,thelistofthe-%oremsthathavegrownoutofsuchexperimentswilxlenlarge.Thenextphase%of;thisdevelopment,inwhichthequestionsthatareconsideredinteresting%arein uencedbycomputationandwherethesequestionsmakecontactwith%therealworld,isjustbeginningtounfold.Isuspectthatultimatelythere%wilxlWbeasizableappliedwingtocommutativealgebra,whichnowexistsin%embryonicN;-ō>;궾1.%NFfoundationsvFDerN;164`1.F:oundations-ō>;궲areanimpGortanttoolforactuallycomputing,sincetheyenableustowrite pGolynomials inawell-de nedwaywhichcanthenbGeimplementedonacom-puter."AfterV=orderingthetermsinpGolynomialsortuplesofpolynomialscom-pletely*,KtheirleadingtermscanbGesingledout.Section1.5showshowtousethoseJ&leadingtermstobuildleadingtermidealsandmoGdules.Conceptually*,thesearesimplerob8jectstohandlethantheoriginalidealsormoGdules.F*orinstance,dthemainresultofSection1.5isMacaulay'sBasisTheoremwhichdescribGesWabasisofaquotientmoduleintermsofacertainleadingtermmoGdule."AdrawbackofMacaulay'sBasisTheoremisthatitneithersayshowtocompute\suchabasisnorhowtorepresenttheresidueclasses.A\Z rstat-temptotoovercomeothesedicultiesismadeinSection1.6wherethereaderisPinstructedonhowtopGerformadivisionwithremainderfortuplesofpoly-nomials. 2ThisproGcedureiscalledtheDivisionAlgorithmandgeneralizesthewell-knownUUalgorithmforunivqariatepGolynomials."However,ƙweshallseethattheDivisionAlgorithmfailstocompletelysolve=theproblemofcomputinginresidueclassmoGdules.NewforceshavetoTbGebroughtintoplay*.Section1.7,theclosingsectionofthe rstchapter,servesMasapreparationforfurtheradvqances.Itisdevotedtoaccumulatingnew^knowledgeandtoenlargingthereader'sbackground.Moreprecisely*,veryBgeneralnotionsofgradingsaredescribGedthere.TheycanbeusedtoovercomeFsomeofthedicultiesencounteredinChapter1.ThisgoalwillbGetheUUtopicofsubsequentchapters.nݠWō>;1.1P9olynomialTRingsٮ17-ō>;1.1**PolynomialRingsq\EvenN=numbGershavetheextrapropGertythateverynon-zeroelementisin-vertible,dandthattheyarecalled elds.AlsoallsquarematricesofagivensizewithentriesinaringformaringwithrespGecttotheusualoperationsofNcompGonentwisesumandrow-by-columnproGduct,but,incontrasttothepreviouslymentionedrings,thepropGertyAgBK=<ڵBkAfails,i.e.theyformaUUnon-commutativering."Althoughj'weshallusematricesintensively*,ourbasicob8jectsarepGoly-nomialringsina nitenumbGerofindeterminatesover elds.SincetheyarecommutativeUUrings,letus rstde netheseob8jects."Recall߰thatamonoidisaset߰S,togetherwithanopGerationS$MSZ w!S궲which7isassoGciativeandforwhichthereexistsanidentityelement,i.e.anelementy1S2S gsuchythat1S@Q9s=s1S=syڲforyalls2S.Whenitisclearwhichmonoidisconsidered,wesimplewrite1insteadof1S.F*urthermore,a=groupisamonoidinwhicheveryelementisinvertible,i.e.suchthatforall sK2S8there existsanelements^0.2KS8which satis essns^0.=Ks^0<>s=1S.AUUmonoidiscalledcomm9utativeUUifUUs8s^0Q=s^0sforUUalls;s^0Q2S.mDe nitionT1.1.1.is~Byaring(R;+;)(orsimplyRYifnoambiguitycanarise)weshallalwaysmeanacomm9utativeղringwithidentityelement,i.e.ΏasetΏRVtogetherwithtwoΏassoGciativeopGerationsΏ+;;ʲ:RH"4[RO!R궲such`that`(R;+)isacommutativegroupwithidentityelement0,suchthat(R>n wf0g;)isacommutativemonoidwithidentityelement1Rb,andsuchthatthe4distributivelawsaresatis ed.Ifnoambiguityarises,weuse41insteadofUU1Rb.UUA eldK qisaringsuchthat(Kn8f0g;)isagroup."F*ortherestofthissection,weletRgbGearing.SomeelementsofaringhavespGecialproperties.F*orinstance,ifr52Rʲsatis esr^ib=0forsomei0,w Wō>;184`1.F:oundations-ō>;궲then{lr‰is{lcalledanilpQoten9t!element,{landifrGr^0=0impliesr^0=0for{lall 궵rG^0n2RDz,athenarA~iscalledanon-zeroQdivisor.AJringwhosenon-zeroelementsareX+non-zeroGdivisorsiscalledanin9tegralؗdomain.F*orexample,every eldisUUanintegraldomain."ThefollowingexampleisnotcentraltothethemesofthisbGook,butitcontributesUUtoshowtheabundanceofrings.'ExampleT1.1.2.c+rG^0V)='(rG)+'(r^0V)s^and%'(rLz/rG^0V)='(rG)'(r^0V),i.e.if'preservestheringopGerations.Inthiscase%weUUalsocallUUSanR-algebraUUwithstructuralThomomorphismUU'.?b)%Given,two,RDz-algebrasSpandTrwhosestructuralhomomorphismsare%'3:R 0p!Sand3 \ :R!Tc,3aringhomomorphism3%3:S 6!Tt²is%calledanR-algebra}homomorphismifwehave%('(rG)Vs)= [ٲ(r)V%(s)%forUUallUUr52Riandalls2S."F*orinstance,goingbacktoExample1.1.2,weseethattheinclusionoftheconstantFfunctionsintoFCW(R)makesCW(R)anR-algebra,Fandthatthemap궵'6 :CW(R) R!Rde nedby/'(f)=f(0)/isaringhomomorphismandalsoanRR-algebraRhomomorphism.F*oreveryringRDz,thereexistsaringhomo-morphism̖':Z!R]which̖maps1qymsbm7Z>to1Rb.Itiscalledthec9haracteristichomomorphismUUofUURDz."Sometimesa eldandagrouparetiedtogetherbyanopGerationofthe eld‰onthegrouptoproGducetheverywellknownalgebraicstructureofaUve}'ctorGMspace.Inthiscasetheelementsofthe eldarecalledscalars,theelementsofthegrouparecalledve}'ctors,andtheopGerationiscalledscalarmultiplic}'ation.UUThoseconceptsgeneralizeinthefollowingway*.De nitionT1.1.4.is~AnIR-moQduleM`isIacommutativeIgroupI(M;+)withanopGerationó:RQMθ!M(calledscalar4Fm9ultiplication)suchthat1oom=mforallm2M,andsuchthattheassoGciativeanddistributivelawsaresatis ed.AcommutativesubgroupܵN2iNMiscalledanR-submoQdule궲ifWwehaveWRN8:qNN.IfNMnɲthenitiscalledapropQersubmoGdule.An궵RDz-submoGduleUUoftheUUR-moGduleRiisUUcalledanidealofUUR."GiventwoԵRDz-moGdulesMandN,amap'd:M{ &!NiscalledanԵR-moQduleyhomomorphism33oran33R-linearmapif33'(mȲ+m^09)8='(m)+궵'(m^09)UUand'(r8m)=r8'(m)UUforUUallr52Riandallm;m^0Q2M.ϠWō>;1.1P9olynomialTRingsٮ19-ō>;"궲Usingothisterminology*,wecansaythatanoRDz-algebraisaringwithan extraand1|s;:::;n `2.*Inthiscasewe%write8M3=hmRj2i.8Theemptysetisasystemofgeneratorsofthe%zeroUUmoGduleUUf0g.?b)%Thep%moGdulep%M@iscalled nitely)generatedifithasa nitesystemof%generators.IfյMisgeneratedbyasingleelement,itiscalledcyclic.A%cyclicUUidealiscalledaprincipalTideal.\Ic)%A9systemofgeneratorsfm jŵ2giscalledanR-basisofMif%every[elementof[MvhasauniquerepresentationasabGove.If[Mvhasan%RDz-basis,UUitiscalledafreeTR-moQdule.?d)%IfIMdisIa nitelygeneratedfreeRDz-moGduleandfm1|s;:::;mrmgisanRDz-%basis&xof&xM,thenrmiscalledtherankofM=anddenotedbyrk XA(M).W*e%remindthereaderthatitisknownthatallbasesofa nitelygenerated%freeUUmoGdulehaveUUthesamelength.HencetherankofUUMlpiswell-de ned.ExampleT1.1.6.c;204`1.F:oundations-ō>;"궲The6/followingnotiongeneralizesthevectorspaceof6/n-tuplesofelements inUUa eld.XDe nitionT1.1.7.is~F*orny2N,thesetRǟ^nG=yf(r1|s;:::;rnq~)jr1;:::;rnꀸ2RǸg궲ofallŵn-tuplesisafreeRDz-moGdulewithrespecttocomponentwiseadditionand9scalarmultiplication.F*or9i>=1;:::;n,9letthetupleei $bGegivenby궵ei5n="(0;:::;0;1;0;:::;0),dwith1oGccurringinthedi^th oposition.Thentheset궸fe1|s;:::;enq~gUUisUUanRDz-basisofRǟ^nE.W*ecallitthecanonicalTbasisofRǟ^nE."Now*werecallthenotionofaunivqariatepGolynomialring.Sinceweshalluseittode nemultivqariatepGolynomialringsrecursively*,westartwithanarbitraryX ringX RDz.W*econsiderthesetRǟ^(N): ofallsequences(r0|s;r1;::: UO)of{Velements{Vr0|s;r1;:::2[Rsuch{Vthatwehave{Vri 6=[0for{Vonly nitelymanyVindicesVis0.UsingcompGonentwiseadditionandscalarmultiplica-tion,EZthissetbGecomesafreeEZRDz-modulewithEZRDz-basisfeijjWi2Ng,EZwhere궵ei=_j(0;:::;0;1;0;0;::: UO)with1oGccurringinpositioniuβ+1.Everyelementof4thissethasauniquerepresentation4(r0|s;r1;::: UO)=Pri2NӵriTLeiTL.4GiventwoelementsUUP㐟i2NTHriTLeiandUUP㐟i2NsieiTL,UUwede ne]>(X +ݴi2NqriTLei)8(X +ݴi2NsiTLei)=X +\i2NZ㉫0 @!rIi X tjg=0+rj6sijZ J1 JAP@eic궲Canyouimaginewherethisstrangerulecomesfrom?(TheanswertothisquestionUUisgivenafterthenextde nition.)"ItViseasytocheckVthatthesetVRǟ^(N) ,togetherwithcompGonentwiseVaddi-tionZandtheproGductde nedabove,ZisacommutativeZringwithidentityZe0|s,andthateiK=e^il1 VSforalli2N.F*urthermore,themapRƸ!Rǟ^(N)޲givenby궵r57!r8e0ȲisUUaninjectiveringhomomorphism.De nitionT1.1.8.is~W*efletfRzŲbGearingandequipRǟ^(N)Hwiththeringstructurede nedUUabGove.a)%IfOweletOx=e1|s,OtheringRǟ^(N)MiscalledthepQolynomialnAringinthe%indeterminate xo9ver R9 and%Bisdenotedby%BRDz[x].Itisacommutative%ring%{andeveryelementof%{RDz[x]has%{auniquerepresentation%{Pi2N$nriTLx^i%withUUrid2Riandri6=0forUUonly nitelymanyindicesUUi2N.?b)%F*or5n1,5werecursivelyde ne5RDz[x1|s;:::;xnq~]=(R[x1|s;:::;xn1])[xnq~]%andUUcallitthepQolynomialTringinTnindeterminatesUUoverUURDz.\Ic)%TheelementsofapGolynomialringarecalledpQolynomials.Polynomials%inOXoneindeterminateareoftencalleduniv\rariatepQolynomials,while%pGolynomialsLinseveralindeterminatesarecalledm9ultiv\rariate˳pQolyno-%mials."Noticegthat,giventhisde nition,themultiplicationoftwounivqariatepGolynomials8P i2N7ֵriTLx^i/and8P i2Nsix^i/comes8outtobGe8P i2N.(Pލ ;i% ;jg=0rj6sij J)x^i,andgthiscorrespGondsexactlytowhatwelearninhighschoGol.ManypropGer-ties8ofaringareinheritedbypGolynomialringsoverit.SomeinstancesofthisgeneralUUphenomenonaregivenbythefollowingpropGosition.Wō>;1.1P9olynomialTRingsٮ21-ō>;PropQositionT1.1.9.qL}'etRbeaninte}'graldomain.~oa)%TheunitsinRDz[x1|s;:::;xnq~]ar}'etheunitsinR. Hb)%Thep}'olynomialringRDz[x1|s;:::;xnq~]isanintegraldomain.Pr}'oof.6Since"MRDz[x1|s;:::;xnq~]was"Mde nedrecursively*,itsucestoprovetheclaimseforen`=1.Giventwoelementsef=`P oi2NSriTLx^ifandg=9=`P ojg2N³r^G0;Zj6x^jƲin궵RDz[x]|"nf0g,9welet9do>=max fi2NjriÊ6=0gande=max fjɸ2Njr^G0;Zj6=0g.Thenthede nitionofthemultiplicationin˵RDz[x]impliesthatoneofthesummandsintherepresentationoftheelementfgdyisrdr^G0፴eKKx^d+ev6=0.F*romthisUUremarkbGothclaimsfollowimmediately*.ҍ"궲BothstatementsofthispropGositionfailifRisnotanintegraldomain.F*or%instance,if%R߲=Z=(4),then(1s+2x)(12x)=1and2xs(2x^2U+2)=0inEnRDz[x].EnF*ollowingtherecursivede nition,anexampleofapGolynomialinthreeUUindeterminatesoverUUQis궵f(x1|s;x2;x3)0=b #( 33233&fes38x31|s)+(x41)x2b J+b7(1x51)x42S( 33333&fes7bٵx1+7x31|s)x52#ݍ*+(x111x)x72|sb ʵx3S+8b7( 1m133&fe12 _Lx1813x21+ jM7l&fe67 ,x31|s)+(x1 jM3l&fe22 ,x31|s)x2+(4x1|s)x22b ʵx23)`+bWx22S+8(4 jM8l&fe13 ,x91|s)x32b ʵx33"궲ManyeZparentheseshavetobGeusedtorepresentmultivqariatepGolynomials inthisway*.ItsureloGoksugly,doGesn'tit?ButwecandomuchbGetter.TheassoGciativeanddistributivelawsprovideuswithamorecompactrepresenta-tion.Infact,everypGolynomialfڧ2RDz[x1|s;:::;xnq~]hasauniquerepresentationofUUtheform fڧ==X + 2Nnҵc t  궲whereoѵ =;( 1|s;:::; nq~)andt =x: 1l1 Dx^ n፴n N,oandwhereonly nitelymanyelementsGc y2R[areGdi erentfromzero.F*orinstance,thepGolynomialabovecanUUbGewrittenas궵f(x1|s;x2;x3)0=x111xx72|sx3S jM8l&fe13 ,x91x32x338x51x42x387x31x52x3 l3l&fes7x1x52x3Y꽸 3۟&fe22 x31|sx2x23S+8x41x2+ jM7l&fe67 ,x31x238x1x22x23+8x42x3+85x22x33Y꽸13x21|sx23S+8x1x2x23+84x22x238x31+ jM1l&fe12 ,x1x23+ l2l&fes3궲Itisimmediatelyclearthattherearemanydi erentwaysofwritingdown thispGolynomialdependingontheorderingoftheelements឵x^11l1xx^7l2|sx3|s,x^9l1x^2l2x^3l3|s,궵x^5l1|sx^4l2x3|s,UUetc."Moregenerally*,letϵrU81,andletMS=8(RDz[x1|s;:::;xnq~])^r LqbGethe nitelygeneratedfreeRDz[x1|s;:::;xnq~]-moGdulewithcanonicalbasisfe1|s;:::;ermgsuchthatpeiW8=(0;:::;0;1;0;:::;0)paspinDe nition1.1.7.Theneveryelement궵m2MlphasUUauniquerepresentationoftheform lm=(f1|s;:::;frm)= ,r X tմi=1X +㉴ 2Nn)Cc ;i .t ei̠Wō>;224`1.F:oundations-ō>;궲whereܵf1|s;:::;fr 2RDz[x1;:::;xnq~],andwhereonly nitelymanyelements 궵c ;i F2RiareUUdi erentfromzero."Intheserepresentations,weusedpGolynomialsoftheformҵx: 1l1 Dx^ n፴n N,where 1|s;:::; n <۸2]N.SincesuchelementswilloGccurfrequently*,wegivethemUUaname.De nitionT1.1.10.o3|LetUUn1.a)%AޛpGolynomial޾f2RDz[x1|s;:::;xnq~]of޾theform޾f=x: 1l1 Dx^ n፴ny suchthat%( 1|s;:::; nq~)#2N^n giscalledatermorpQo9werproduct.Thesetofall%termsUUofUURDz[x1|s;:::;xnq~]isdenotedbyT^n ӲorT(x1|s;:::;xnq~).?b)%F*oratermt)=x: 1l1 Dx^ n፴n Q2T^nq~,thenumbGerdeg(t))= 1~+` j+` n is%calledUUthedegreeofUUt.\Ic)%ThepRmappRlogO:T^ne!N^n вde nedbypRx: 1l1 Dx^ n፴n a7!( 1|s;:::; nq~)ispRcalled%theUUlogarithm.?d)%Ifr1g1andM=(RDz[x1|s;:::;xnq~])^r >isthe nitelygeneratedfree%RDz[x1|s;:::;xnq~]-moGduleLwithcanonicalbasisLfe1;:::;ermg,thenaterm%ofzMiszanelementoftheformteiZƲsuchthattT2T^n wand1irG.%TheNsetofalltermsofNMewillbGedenotedbyT^nq~he1|s;:::;ermiorby%T(x1|s;:::;xnq~)he1;:::;ermi."ThedqsetdqT^nisacommutativedqmonoid.Itsidentitydqelementisdq1=x^0l1'|jx^0፴nq~.ThemonoidT^n wdoGesnotdependontheringofcoecientsRDz.ThesetT^nq~he1|s;:::;ermi|can|bGeconsideredasthedisjointunionofröcopiesofT^n\wherethe&symbGols&e1|s;:::;ersimplyindicatewhichcopyof&T^n aweareconsidering.De nitionT1.1.11.o3|Let õn$1, letf-=P_ 2Nn%_c t 2RDz[x1|s;:::;xnq~]bGe apGolynomial,UUandletUUm=Pލ USr% USi=1tJP' 2Nn=?c ;i .t eid2M3=(RDz[x1|s;:::;xnq~])^rm.a)%F*oreveryȵ O2N^nq~,i2f1;:::;rGg,theelementȵc ;i2յRiscalledthe%coQecien9tUUofthetermUUt eiinm.?b)%The set ft eiJ32T^nq~he1|s;:::;ermijc ;i6=0gis calledthesuppQortofm%andUUdenotedbyUUSupp=(m).\Ic)%If-fڧ6=0,-thenumbGermaxfdeg#(t )jt y2Supp(f)gis-calledthedegree%ofUUfhandUUdenotedbydegx(f)."F*or^example,thesuppGortofthepolynomial^ݵfꊸ2Q[x1|s;x2;x3]^abGove^con-sistsof17terms,andthesequenceoftheirdegreesis19;15;10;9;7;6;5;5;5;궲5;5;4;4;4;3;3;0.sW*ehavesorderedthetermsinsSupp[(f)sbydecreasingde-gree.CHowever,thisisnotenoughtoorderthemcompletelysincetherearesev-eral8termswiththesamedegree.Completeorderingson8T^naandT^nq~he1|s;:::;ermi궲willUUbGeexaminedinSection1.4."TheүpGolynomialringcanbeusedtode neinterestingringhomomor-phisms.wOneofitsfundamentalpropGerties,calledtheUniversalPropGerty*,saysEthatringhomomorphismsstartingfromapGolynomialringareuniquelyde ned3bytheimagesoftheindeterminates,andthoseimagesmaybGechosenfreely*.׉Wō>;1.1P9olynomialTRingsٮ23-ō>;PropQositionT1.1.12.w(Univ9ersalTPropQertyofthePolynomialRing) L}'et$S be$anR-algebra$withstructur}'alhomomorphism':R߸!S,$letn1,andlets1|s;:::;sn Qbb}'eelementsinS.Thenther}'eexistsauniqueringhomo-morphismw X:cRDz[x1|s;:::;xnq~]!S suchwthat [ٸjR ='and [ٲ(xiTL)=sifor궵i=1;:::;n.Pr}'oof.6Byinduction,itsucestoprovetheclaimforߵn=1.F*ord0and궵c0|s;:::;cdpʸ2RDz,Q;weletQ; [ٲ(Pލ ;d% ;i=02ciTLx^il1)=Pލ USd% USi=0tJ'(ciTL)s^il1|s.Q;ItiseasytocheckQ;thatthisNde nesaringhomomorphismhavingtherequiredpropGerties.Ontheotherhand,since βhastobGecompatiblewithadditionandmultiplication,thisUUde nitionisforcedupGonusandUU .isuniquelydetermined.0M="궲ABringahomomorphisma 4:de nedinthiswayaisalsocalledanev\raluationhomomorphism,=andtheimageofapGolynomial=fQviscalledtheev\ralua-tiondf(s1|s;:::;snq~)offwat(s1;:::;snq~).dInthespGecialcasewhendSs1=ߤRDz,anevqaluationhomomorphismǵ  :ֵRDz[x1|s;:::;xnq~] L!Risalsocalledasubsti-tutionn|homomorphism.Usingevqaluations,wecanspGeakaboutgeneratorsofUURDz-algebrasUUinthefollowingmanner.De nitionT1.1.13.o3|LetUUSbGeUUanRDz-algebra.a)%A-bset-ܸfs ڸjڠ2gof-elementsofSiiscalledasystemofgen-%erators̲of̵S~Yifforeveryelement̵s2S~Ythereisa nitesubset%f1|s;:::;tVg- ofand- apGolynomial- f(x1;:::;xtV).2RDz[x1;:::;xtV]- such%thatUUs=f(s1 %Z;:::;st).?b)%TheRDz-algebraS?0iscalled nitely8generatedifithasa nitesystem%ofUUgenerators.CorollaryT1.1.14.lAn$R-algebr}'aSis$ nitelygeneratedifandonlyifther}'e*existsanumber*n{2Nand*asurje}'ctiveR-algebra*homomorphism궵':RDz[x1|s;:::;xnq~]!S."InΑotherwor}'ds,every nitelygeneratedΑR-algebraSbisΑoftheform궵ST͍Z+3Z= ۵RDz[x1|s;:::;xnq~]=I\wher}'eIisanidealinRDz[x1|s;:::;xnq~].Pr}'oof.6Thisfollowsfromthefactthatasetfs1|s;:::;snq~gofelementsofS궲iswasystemofgeneratorsofwS[ifandonlyiftheRDz-algebrahomomorphism궵':RDz[x1|s;:::;xnq~]!Sde nedUUbyUUxid7v!sifori=1;:::;nisUUsurjective.X D"궲F*ortantRDz-algebraSwhichhasa nitesystemofgeneratorsfs1|s;:::;snq~g,theccorrespGondingisomorphismcST͍e+3e=b[RDz[x1|s;:::;xnq~]=IEiscalledapresen9ta-tion"of"Sbygeneratorsandrelations,andtheidealIIiscalledtheidealofalgebraicTrelationsUUamongUUfs1|s;:::;snq~gwithcoGecientsinRDz.Wō>;244`1.F:oundations-ō>;%Exercise\M1.LLet$5" cmmi9dF& cmsy92,a msbm9ZbAeanon-squaren9umber,andletK=FQ[SpRS׉aH)d ?], %where?w9euse?Sp oS׉aH)dQ"=iapaH AdRif?d<0.?Pro9vethat?Kisa eld,andthatevery .%elemen9tzr|2;%Khaszauniquerepresentationzr|=;%aQ+bSpRS׉aH)d2witha;b;%2Q.%The eldKiscalledthe.t : cmbx9quadraticnCumbKer eldgeneratedb9ySp SS׉aH)d@. %After)&represen9ting)&r;s2Kb9ypairsofrationals,giveformulaefor)&r]5+ns,%rA,TrO8s,TandP Il#Aacmr61HaHp0%;cmmi6r=~forrӍ6=0.%Exercise2.0Sho9w~that,uptoauniqueisomorphism,thepAolynomial%ring 5R>[x1*;::: ;xn7]is 5theonly 5R>-algebrasatisfyingtheuniv9ersalpropAerty%statedinPropAosition1.1.12.Inotherw9ords,supposethatTisanother%R>-algebra#togetherwithelemen9ts#t1*;::: ;tnV2TH,suchthatwheneveryou%ha9veanR>-algebraS#togetherwithelemen9tss1*;::: ;sn2zvS,thenthere%existsvauniquevR>-algebrahomomorphism p:Tک!Ssatisfying R(ti,r)=si%fori=1;::: ;n.Thensho9wthatthereisauniqueR>-algebraisomorphism%R>[x1*;::: ;xn7]!T^7suc9hTthatTxi87!tiAfori=1;::: ;n.%Exercise^3. Sho9wthatthemaplog:JQT-=nh!N-=n Eisanisomorphismof%monoids.%Exercise4.eLetv1RV='(a11N;a21;::: ;an1);:::;vn E='(a1n;a2n;::: ;ann n)%bAeelemen9tsofZ-=n7,andletA=(aijM)2Matn)(Z)bAethematrixwhose%columns!arethecoAordinatesof!v1*;::: ;vn7.Sho9wthatthesetfv1*;::: ;vn7g%isTaTZ-basisofZ-=n 3ifandonlyifdet(A)2f1;1g.%ExerciseT5.jLetTSYbAeTthesetoffunctionsfromZtoZ. *?a)7xSho9wYthatYS^withtheusualsumandproAductoffunctionsisaZ-7xalgebra.)b)7xUseS[considerationsabAoutthecardinalit9yofS[S`toshowthatS[S`isnot7xaT nitelygeneratedTZ-algebra.%Exercise6.LetRbAearingandI.anon-zeroidealofR>.Pro9vethatI%is&6afree&6R>-moAduleifandonlyifitisaprincipalidealgeneratedb9ya%non-zeroAdivisor.%Exercise7.LetZRbAeZaring.Sho9wthatthefollowingconditionsare%equiv|ralen9t. *?a)7xTheTringTR%isa eld.)b)7xEv9eryT nitelygeneratedTR>-moAduleisfree.*6c)7xEv9eryTcyclicTR>-moAduleisfree.%Exercise8.Let3K׋bAe3a eld,P -=JK[x1*;x2],3andIbAetheidealinP%generatedTb9yTfx1*;x2g.Sho9wthatTIɯisnotafreePH-moAdule.1T utorialT1:P9olynomialRepresentationI궲In|whatfollowsweworkoverthering|K[x;y[ٲ],whereKAisoneofthe elds de ned]inC0oLCoA .UsingDe nition1.1.8,weseethatwecanrepresenteverypGolynomial;fڧ2K[x;y[ٲ]as;alistoflists,whereaunivqariatepolynomial;a0+궵a1|sx.U+Q+adx^dsuchPthatPa0;:::;adpʸ2K+andad6=0isPrepresentedbythelistUU[a0|s;:::;ad].̠Wō>;1.1P9olynomialTRingsٮ25-ō>;"궲TheGpurpGoseofthistutorialistoprogramthetransitionfrompolynomials toklists(andback),andtoseehowadditionandmultiplicationofpGolynomialscanbGecarriedoutusingtheirlistrepresentations.ThusthistutorialismainlyintendedasanintroGductiontothekindofC0oLCoA"programmingweaskyoutoUUdoinothertutorials."TherearesolutionsofpartsofthistutorialinAppGendixC.1.Sincemostzprogramsrequiretheuseoflists,wesuggestyoureadAppGendicesA.6and}OB.5bGeforeyoustart.A}E(long)listofC0oLCoA"commandsfordealingwithlists+canbGegeneratedbyinvokingtheon-linemanualwiththecommandH.Commands('list');a)%W*riteMaC0oLCoA"programMReprUniv.5(::: )Mwhichtakestwoarguments:the% rst3x^2x+1andf2\v=y[ٟ^2+2y+3tolistsand%back.%Hint:A]simpleFor-loGoporthesumoveranappropriatelyconstructed%listUUwilldothetrick.\Ic)%W*riteYC0oLCoA!functionsYAddUniv)Ӳ(::: )YandMultUniv.Yв(:::)whichYtaketwo%listsbrepresentingunivqariatepGolynomialsandcomputethelistsrepresent-%ing8theirsumandproGduct,respectively*.Usetheprogram8ListToPoly7Ų(::: )%toUUcheckthecorrectnessofbGothfunctions.%Hint:`Whenimplementingthesum,youshouldswitchthesummands%suchthatthe rstonehaslargerdegree(i.e.alongerlist).Thenyoucan%addUUtheelementsofthesecondlistontothe rstone.%F*or3theimplementationoftheproGduct,youmaywanttoconsiderthe%formula(a0B+ 6+ adx^d)(b0B+6+beKKx^e)=Pލ USd+e% USi=0(Pލ ;i% ;jg=0aj6bij J)x^iTL.It%mayQbGeusefultobringbothliststothesamelength rst(byappending%zeros).TheinnersumcouldbGerealizedbyaconstructionlikeSum(L)",%whereUUListhelistofallUUaj6bijJ.?d)%W*rite~VaC0oLCoA$!program~VReprPoly/~>(::: )~VwhichrepresentsapGolynomial%fg2صK[x;y[ٲ]asalistoflistsofelementsofK.Theelementsofthebig%listarelistsrepresentingunivqariatepGolynomialsinܵK[x],namelytheco-%ecientsofthedi erentpGowersofyYiinthepGolynomial,consideredasan%elementsofs(K[x])[y[ٲ]asinDe nition1.1.8.b.F*orinstance,thepGolynomial%f1C=x^2S+82xy+3y[ٟ^2 -isUUrepresentedbythelistoflistsUU[[0;0;1];[0;2];[3]].Wō>;264`1.F:oundations-ō>;%Hint:TheC0oLCoA![commandDeg(F,x)returnsthedegreeofapGolynomial %F1withڕrespGecttotheindeterminatex.ThefunctionڕShorten*(::: )ڕin%AppGendixC.1removestrailingzerosfromalist.Thesefactsandadouble%For-loGopxzaregoodenoughfora rstsolution.Moreelegantly*,youcanalso%useUUReversed(Coefficients(::: ))QandUUacleverlistconstruction.\Ie)%W*riteaC0oLCoA b,UUinterchangeaandb.)3)7xComputearepresentationbi=q[az+r-withqӸ2iNandaremainder7x0 KrRh;1.1P9olynomialTRingsٮ27-ōWӍ)2)7xF*orm8thetriples8(c0|s;d0;e0)=(&h33jaj33ʉfe p`a |ֵ;0;jaj)8and(c1|s;d1;e1)=(0;&h۷jbj۟ʉfeDD`b UR;jbj). )3)7xCheckwhethere1Ce0|s.Ifthisisnotthecase,interchange(c0|s;d0;e0)7xandUU(c1|s;d1;e1).)4)7xW*ritee0 Znintheforme0'I=ֵq[e1l+rG,whereq2Nand0r1UUbGeUUaprimenumber."(Note:Sever}'alpartsrequiresomebasicknowledgeof eldtheory.)a)%Let߆KbGe߆a nite eldofcharacteristicp.ShowthatthenumbGer߆q;_of%elementsof痵KisapGowerof痵p,i.e.thereisanumbGer痵e>0suchthat%q"=p^eKK.UU(Hint:NotethatUUK qisaZ=(p)-vectorspace.)?b)%LetLbGeanalgebraicallyclosed eldofcharacteristicp.Provethatthere%existsauniquesub eldFqnofLwhichhasq_elements,andthatitisthe%setUUofroGotsoftheequationUUx^qu8x=0.\Ic)%Showthatevery eldKӲwithqQelementsisisomorphictoFqj,thatthere%isanirreduciblepGolynomialf2ofdegreeeinZ=(p)[x],andthatthereis%anUUisomorphismUUKT͍~4+3~4= jZ=(p)[x]=(f).%Hint:ͅF*orthesecondpart,usethefactthatthemultiplicativeͅgroup%Kn8f0gUUisUUcyclic.5cWō>;284`1.F:oundations-ō>;?ֲd)%ImplementUoaC0oLCoA!functionUoIrredPoly3T(::: )Uowhichcomputesthelistof %alltmonicirreduciblepGolynomialstfofdegreed=degOfڧeinZ=(p)[x],%i.e.8whosecoGecientof8x^dgis1.8Proceeddegreebydegree,startingwith%[x;xC 1;:::;xp+1]danddappGendingthelistofallmonicpolynomials%ofAdegreeAdwhicharenotdivisiblebyoneoftheirreduciblepGolynomials%ofUUdegreeUUd=2.\Ie)%Using;fڧ=Last(IrredPolyJꀲ(::: )),;wecanrepresenteveryelement;r52K%asalistr݃=f[r1|s;:::;reKK]ofelementsr1;:::;re ᱸ2fZ=(p)suchthat%r?+w(f)=r1t+r2|sx+ +reKKx^e16+(f).W*riteC0oLCoA 5functionsFFAdd(::: ),%FFNeg?(::: ),Y2FFMult# (:::),Y2andY2FFInv#(:::)Y2whichcomputethelistsrepre-%sentingsthesums,negatives,proGducts,andinversesofelementsofsƵK,%respGectively*.UU(Hint:UsethebaseringS::=Z/(P)[x].)f)%ComputeUUarepresentationofthe eldUUF16 ;anditsmultiplicationtable.G0Wō>;1.2UniqueTF:actorizationٮ29-ō>;1.2**UniqueFactorizationF$"EverythingN;304`1.F:oundations-ō>;Pr}'oof.6The onlynon-trivialimplicationistoshowthatif fisirreducible, thenitisaprime.W*ehavealreadymentioned(seeExample1.1.6)thatif궵K/@isx$a eld,theneveryidealinx$K[x]isx$principal.SuppGosex$fisirreducible,궵fޱj"ab,WandWf-a.WThenWab=g[fkPforsomeWg&2K[x].SincetheidealW(a;f)isgenerated0#byadivisorof0#f,wehave0#(a;f)=(1),0#andtherefore1=rGa|+sf궲for7some7r;s2K[x].Thusweget7b=rGabr+sbfڧ=rg[f+sbfK-and7fڧjb. "궲In Exercise9wewillseeanelementofanintegraldomainwhichisir-reduciblebutnotprime.Thefollowingexampleshowsthatthenotionofirreducibility(inapGolynomialringdependsstronglyonthe eldofcoe-cients.4ExampleT1.2.3.c;1.2UniqueTF:actorizationٮ31-ō>;"궲As[Uwithintegers,itispGossibletode negreatestcommondivisorsand leastUUcommonmultiplesinafactorialdomain.ȍDe nitionT1.2.6.is~LetzRAbGezafactorialdomain.W*esaythattwoirreducibleelementsK&ofK&R^areassoQciatediftheydi eronlybymultiplicationwithaunitofRDz.LetthesetPˠRnbGeobtainedbypickingoneelementineachclassNofassoGciatedirreducibleelementsofNRDz.F*urthermore,letm궸2and궵f1|s;:::;fm _2RLn8f0g.a)%Let2nf1C=c1'Qp2P!np^ p$andf2=c2'Qp2P!np^ p nbGe2nfactorizationsof2nf1and%f2/with`units`c1|s;c2V2RDz,with pR; pyl2N,andwith pyl= p=0`for`all%butUU nitelymanyUUp2P}.UUThentheelementȍ#gcd#(f1|s;f2)=寧Q7p2P4pmin @f p2Դ; pg$7%isUUcalledagreatestTcommondivisorUUofUUf1Ȳandf2|s,andtheelementlcm(f1|s;f2)=寧Q7p2P4pmax̟f p2Դ; pg%isUUcalledaleastTcommonm9ultipleUUofUUf1Ȳandf2|s. ?b)%Ifggcdi(f1|s;f2)=1,gwesaythatgf1|s;f2cڲarecoprimeorrelativ9elyVprime.\Ic)%F*or7m8<>2,7wede neagreatest#ecommondivisor7andaleast#ecom-%monTm9ultipleUUofUUf1|s;:::;fm recursivelybyȍ`gcdo(f1|s;:::;fm)=gcd(gcd(f1;:::;fm1);fm)_lcmou(f1|s;:::;fm)=lcmUS(lcm;(f1;:::;fm1);fm)!?d)%LetصfD=0cQ qp2P p^ pzwithaunitc02RDz,with pD2N,andwith pD=00 %forȇallbut nitelymanyȇp12PbGeȇthedecompositionofanelement%fڧ2RLn8f0gUUintoUUirreduciblefactors.Thentheelementȍsqfree,D(f)=寧Q7p2P4pmin @f1; p2Էg%isUUcalledasquarefreeTpartofUUf.ȍ"ItVisclearthatthede nitionofgreatestcommondivisorsandleastcom-monoamultiplesdoGesnotdependontheorderoftheelements.Itisalsoclearthatgreatestcommondivisors,leastcommonmultiples,andsquarefreepartsofPelementsPf1|s;:::;fm _2Rڟnf0gchangePonlybyaunitifwechoGoseadi erentsetTofrepresentativesT޸P'[forTtheequivqalenceclassesofirreducibleelements.W*eshallGthereforespGeakofthegreatestcommondivisorandtheleastcommonmultiple(of(f1|s;:::;fm _2RPnf0g,aswellasthesquarefreepartof(fڧ2RPnf0g,whileUUalwayskeepinginmindthattheyareuniqueonlyuptoaunit."Inthefollowing,wedescribGesomeconnectionsbetweengreatestcom-mon divisors,leastcommonmultiples,andidealtheory*.Firstwecharacterizegreatest commondivisorsandleastcommonmultiplesbydivisibilitypropGer-ties. mޠWō>;324`1.F:oundations-ō>;PropQositionT1.2.7.q(CharacterizationTofgcdandlcm) "L}'etRbeafactorialdomain,andletf1|s;:::;fm _2RLn8f0g~oa)%An element ڵf$2Risthegr}'eatest commondivisorof ڵf1|s;:::;fm uifand%onlytiftfBj賵fifori=1;:::;mandteveryelementtgD2R;suchthattgDjfi%fori=1;:::;msatis esg"jf.Hb)%Anelementfڧ2Risthele}'astcommonmultipleoff1|s;:::;fm ifandonly%if'fihj?ffori=1;:::;mand'everyelement'g2Rsuchthat'fihjg2for%i=1;:::;msatis esfڧjg[.Pr}'oof.6FirstInweprovea).F*orIni]=1;:::;m,Inletfi7=ciQpp2P ^Gp^ pibGeInthefactorizationsofsfiTL.Usingthede nitionandinductiononm,weseethatgcd"(f1|s;:::;fm)J=Q p2P.p^min @^f p1;::: ; pm Vg?(8.=Thusitfollowsimmediatelythatgcd"(f1|s;:::;fm)UUdividesfifori=1;:::;m."Nowletg"2RbGeacommondivisoroff1|s;:::;fm,andletg"=cQ qp2P p^ p궲bGe9thefactorizationof9g[ٲ.F*oreveryi<2f1;:::;mg,9theconditiongGjzjgcd>|(f1|s;:::;fm).6Thusweget6gcd6(f1|s;:::;fm)>z2(h)=(f1;:::;fm),6asclaimed. "궲Now)itistimetomovedirectlytotheheartofthissection.W*ewanttoproveuthatpGolynomialringsoveru eldsarefactorial.ThenextlemmaisthekeyUUtothisproGof.!~Wō>;1.2UniqueTF:actorizationٮ33-ō>;De nitionT1.2.9.is~Let RҲbGe afactorialdomainandfڧ2RDz[x]TNnf0g. Agreat- estYcommondivisorofthecoGecientsofY۵fmjiscalledacon9tentY۲off.AsbGefore,'weusuallyspeakofthecontent'of'f;randdenoteitbycontW(f).'Ifcont&*(f)=1,UUwesaythatUUfhisprimitiv9e.4LemmaT1.2.10.bG (Gau'sTLemma)~L}'etRbeafactorialdomain,andletfV;g"2RDz[x]b}'enon-zeropolynomials.~oa)%Wehavecont0[(fg[ٲ)=contc(f)8contT(g[ٲ).Hb)%IffV;gar}'eprimitive,soisfg[.Pr}'oof.6Itisclearthata)followsfromb),sinceeverypGolynomialfwisoftheformfl=Xcont(f)s\q0~f forsomeprimitivepGolynomial\qӽ~f .So,letusproveb).rW*ewriterf=P,i2NriTLx^i andgcʲ=P,i2NsiTLx^i withri;si\=2RDz.rLetp궲bGenanirreducibleelementofnRDz.ThehypGothesisimpliesthatthenumbGers궵j^=+minvٸfi+2Njp-riTLgX`andk²=+minfi+2Njp-siTLgX`exist.X`NowRl'isfactorialandLpisLirreducible,henceprime.AsitdoGesnotdivideLrjandsk됲,Litdoesnot qdivide qrjĸskeither.Thechoiceofjandk\yieldsthatpdoGesnotdividetheCcoGecientofCx^jg+k Uinfg[ٲ.CThereforeitdoesnotdivideCcont(fg[ٲ),andweareUUdone.mLemmaT1.2.11.bG L}'etL3R_beL3afactorialdomain.Theneverynon-zer}'oelementofRDz[x]hasafactorization.Pr}'oof.6Letr&fڧ2RDz[x]rnf0gbGer&anon-unit.Sincefisoftheformfڧ=contc(f)rg궲withaprimitivepGolynomialg[ٲ,andsincecont(f)hasafactorizationbyassumption,UUwemayassumethatUUfhisprimitive."W*e{proGceedbyinductionon{d=deg)(f).{Ifd=0thenf=contu(f)=1hasatrivialfactorization.Ifd>0andfoisirreducible,thereisnothingtoprove.Otherwise,letf%G=g[hwithnon-unitsg;h2RDz[x]Vnf0g.Ifoneofthetwo,KsayKg[ٲ,hasdegreezero,i.e.ifg-2ѱRDz,then1ѱ=contn%(f)=g]cont?(h),contradicting{thefactthat{gisnotaunit.Thusthedegreesof{gandh궲are\bGothstrictlylessthan\d,andanapplicationoftheinductivehypGothesis nishesUUtheproGof.3PropQositionT1.2.12.wL}'et=sRQ:be=safactorialdomain.ThenRDz[x]isalsoafac-torialdomain.Pr}'oof.6AccordingWtoPropGosition1.2.5andLemma1.2.11,wehavetoprovethatHeveryirreduciblepGolynomialinHRDz[x]isHprime.LetHQ(R)bGeHthe eldoffractions>of>RDz.Inordertoprove>theclaim,weshallargueasfollows:if>f궲is&anirreducibleelementof&RDz[x],weshowthatitisirreduciblein&Q(RDz)[x],hence;344`1.F:oundations-ō>;궲element@*r52RSsuch@*thatrGfڧ=g2|sh2withg2;h2C2RDz[x].@*F*romLemma1.2.10 weknowthatrƙ=|cont(g2|s)cont(h2).Thuswecansimplifyandgetanewequation̯f=g3|sh3 I"with̯primitivepGolynomialsg3|s;h3 u2RDz[x].Sincethede-grees36of36g3 andh3are36pGositiveand36RFisanintegraldomain,neitherisaunit,fcontradictingtheirreducibilityofff.Thuswehaveshownthatffz=isirreducibleUUinUUQ(RDz)[x]."TheringصQ(RDz)[x]isaunivqariatepGolynomialringovera eld,hencein궵Q(RDz)[x]MGeveryMGirreducibleelementisprime(seePropGosition1.2.2).Conse-quently*,SthepGolynomialSf isprimeinQ(RDz)[x].Itremainstoshowthatf궲isyprimeasanelementofyRDz[x].W*estartwithanequationefo8=[g[h,where궵e;g[;h2RDz[x].IfwereaditinŵQ(R)[x],wededucethatŵgorhmustbGeamul-tipleoff>inQ(RDz)[x].Assumeforinstancethatg=]q[f>withq2]Q(RDz)[x].Bycclearingthedenominators,wegetcrGg"=pfforsomer52R *andp2RDz[x].Therefore.(wehave.(rlconte(g[ٲ)0v=cont(p),.(andaftercancelling.(ruEweobtain궵g"2(f),UUaswastobGeshown.煍"궲By=repGeatedlyapplyingthepreviousproposition,weseethatpolynomialringsover eldsarefactorialdomains.Thisisoneoftheirfundamentalprop-ertiesUUanddeservestobGethe naltheoremofthepresentsection.,TheoremT1.2.13.jUFL}'et>Kbe>a eldandnK1.>Thenthep}'olynomialring궵K[x1|s;:::;xnq~]isafactorialdomain.-(%ExerciseT1.jPro9veTthatprimeelemen9tsareirreducible.煍%ExerciseO2.Let_p=101._W:riteaC'oZ:CoAn!programwhic9hcheckswhether %aqgiv9enpAolynomialqfq2Z=(p)[x]ofqdegreedegs(f)3isqirreducible.Pro9ve%theTcorrectnessofy9ourmethoAd.%Exercise|3.bLetpbAeaprimen9umberandx*:%Z[x]!Z=(p)[x]the%canonicalThomomorphism. *?a)7xSho9wK thatifK f2OZ[x]isK amonicpAolynomialandK R(f)isK irreducible7xthenTfisTirreducible.)b)7xPro9veTthatstatemen9ta)is,ingeneral,falseifTfisnotmonic.*6c)7xGiv9eTacounterexampletotheconverseofa).%ExerciseP4.9wFindxafactorialdomainxRG6=7eZwhic9hxdoAesnotcontaina% eld.%Exercise)5./Sho9w&thatif&R7=isafactorialdomainand33 eufm9pisminimal%amongTtheprimeidealsdi eren9tfromT(0),thenpisprincipal.%Exercise6.ZLetJRZPbAeJanin9tegraldomainwiththepropert9ythatevery%non-zero 3non-unithasafactorization.Assumethat,forall 3a;b2R4nf0g,%theTidealT(a)8\(b)isTprincipal. *?a)7xPro9veQthat,giv9ennon-assoAciatedirreducibleelementsQa;b2RFn6f0g,7xw9eThaveT(a)8\(b)=(ab).7xHint:Let(a)Y\(b)Nu=(c),letab=rAc,andletc=sa.Sho9wthats7xcannotTbAeaunit.ThendeducethatTrWhastobeaunit.#Wō>;1.2UniqueTF:actorizationٮ35-ō>;)b)7xUse|a)topro9ve|thatt9wo|factorizationsofan9yelementarethesame 7xupTtoorderandunits.*6c)7xConcludeTthatTR%isafactorialdomain. u%Exercise47.ConsiderwtheringwR=VZ[pRaH Ÿ5].Pro9vewthattheelemen9ts u%f1m=2+2pRaH Ÿ5and f2=6do notha9ve agreatestcommondivisorinthe%senseTofPropAosition1.2.7.a.%Hint:TSho9wthatbAothT2and18+p ÊaH Ÿ5areTcommondivisors.%Exercise8.LetRLbAeafactorialdomainanda;bQ2RGn f0gt9woco-%primevkelemen9ts.ProvethatthepAolynomialvkaxe+bisvkanirreducibleelemen9t%ofTR>[x].%Exercise9.8LetFKC.bAeFa eld,Pk=="ZK[x1*;x2;x3;x4],FandFpbAethe%principalTidealgeneratedb9yTfq=x1*x488x2x3*. *?a)7xSho9whthatthepAolynomialhfisirreducibleinPH.DeducethatPU=pis7xanTin9tegraldomain.)b)7xPro9veQthattheresidueclassofQx1moAdulopisQirreducibleinQPU=p,but7xnotTprime.UsethistoinferthatTPU=pisTnotfactorial.%ExerciseG10. LetKbAea eldandK(x)the eldoffractionsofK[x].%W:eTconsidertheringTR=K[x1*;x2]=(x1x2881). *?a)7xSho9wU9thatU9RewisisomorphictoaK-subalgebraofK(x)whic9hcontains7xK[x].(Hint:T:rytomapx1htoxandx2toP )1aH/0x J&.T:osho9wthatonly7xm9ultiples+fof+fx1*x2e:[1areinthek9ernelofthismap,writepAolynomialsin7xK[x1*;x2]as?(zi cmex9P W]i>09x-=ih15 6fi,r(x1*x2)+?P i>0x-=ih25gi,r(x1x2)+cwherec2K.) ()b)7xUsing[a),w9emayassume[K[x]NRK(x).[Inthissituation,sho9w7xthatxisaunitinRandev9eryelementgNz2RcanbAewrittenas7xgp=x-=r:8fwhereTrӍ2Zandfq2K[x].*6c)7xPro9ve(that(R8Visafactorialdomain.(Hint:Usetherepresen9tationgiven7xinTb)tosho9wthateveryirreducibleelementisprime.)%Exercise11.jF:orauniv|rariatepAolynomialf,w9edenoteitsderivhativCe%b9yTf-='q% cmsy60r.TLetK&isa7xproAductTofthreelinearfactors.*6c)7xUseTa)andb)topro9veTthatthefollo9wingconditionsareequiv|ralent.<1)I?qF:orDallDa;b2K,thereisanelemen9tDc2KsuchDthatDa-=2+mb-=2m=c-=2*.<2)I?qF:or@aev9erymonicpAolynomial@af~21K[x]of@adegree3which@aisaI?qproAduct-Eofthreelinearfactors,-Ef-=0ٷisaproductoft9wo-Elinearfac-I?qtors.$ӠWō>;364`1.F:oundations-ō>;T utorialT4:EuclideanDomains궲In]general,itisdiculttodecidewhetheragivenringisfactorial,andcon- sequentlythereexistsonlyaratherlimitedsupplyofexamplesoffactorialdomains.qThepurpGoseofthistutorialistoprovidethereaderwithatoolforconstructingUUordetectingaspGecialkindofnon-trivialfactorialdomains."W*eysaythaty(R;')(orysimplyRDz)isaEuclidean%domainifRisadomain[and['isafunction'ѽ:RPn="f0g!N[suchthatforalla;bѽ2RPn="f0g궲theUUfollowingpropGertieshold.1)%IfUUajbthen'(a)'(b).2)%Therewexistelementswq[;rG2ӵRsuchwthataӲ=q[bO+randweitherrG=0or%'(rG)<'(b)."First8ofall,prove8thatinaEuclideandomain8RLthefollowingadditionalruleUUholds.3)%Let͵a;b2Rnf0g.Ifb=acforsomenon-unitc2RDz,then'(a)<'(b).%Hint:_Use2)andwrite_a=q[b+rG._Showthatr56=0,hence'(rG)<'(b).%DeduceUU(18q[c)a=rG,UUhence'(a)'(rG)."SomeCrings,whichshouldbGefamiliartothereader,areinfactEuclideandomains.a)%ShowFthattheringofintegersFZ,togetherwiththeabsolutevqaluefunc-%tion,UUisaEuclideandomain.?b)%CheckVWthateveryunivqariatepGolynomialringVWK[x]overVWa eldVWK,to-%getherUUwiththedegreefunction,isaEuclideandomain.InUUthefollowing,weletUURibGeaEuclideandomain.\Ic)%ShowEthatifEm=minqƸf'(a)ja2R‹nf0gg,Ethenfa2R‹nf0gj'(a)=mg%isUUthesetofunitsofUURDz.?d)%Use!1),andanargumentsimilartothatgiveninLemma1.2.11,toprove%thatUUeverynon-unitinUURLn8f0ghasUUafactorization.\Ie)%Use+2)toshowthatin+R?mthereisanotionofgcd+(a;b)fora;b2RInf0g%inthesenseofPropGosition1.2.7.a,andthatgcd(a;b)canbeexpressed%asUUrGa8+sbwithr;s2RDz.f)%Usee)toprovethatinRUeveryirreducibleelementisprime.Conclude%thatUURiisUUfactorial.%Hint:=?Let=?pbGeirreducibleandabI=cp.If=?pdoGesnotdividea,then%gcd4(a;p)=1.UUHenceUU1=rGa8+sp,UUandthereforeUUb=q.g)%ConsiderGthesubringGZ[i]=fa'+bija;b2ZgofC.GItiscalledthering%ofUUGauianTn9umbQers.)1)7xLetr':Z[i][nf0g!NbGerde nedby'(a[+bi)=a^2׎+[b^2|s.rShowthat7x'UUmakesZ[i]intoUUaEuclideandomain. 7xHint:òLetõz1=za[Բ+bi;z2=zc+diòandz=hQz1Qʉfe6z2\=&hQ(a+bi)(cdi)Qʉfe.#t :c2 +d26θ2Q[i].7xChoGose%for%q~72"^Z[i]a\goGod"approximationof%zJandwritez1Ѳ=7xq[z2S+8rG.)2)7xFindUUthesetofunitsofUUZ[i].%:Wō>;1.2UniqueTF:actorizationٮ37-ō>;)3)7xLetRp2NbGeRaprimenumber.RShowthatRpisreducibleinZ[i]if 7xandUUonlyifthereexistUUa;b2NsuchUUthatp=a^2S+8b^2|s.7xHint:UUProvethatthemapUU'iscompatiblewithmultiplication.)4)7xShowthatif⳵z72Z[i]issuchthat⳵'(zp)isaprimenumbGerinZ,then7xzisUUprimeinUUZ[i].)5)7xRepresentingelementsofZ[i]aspairsofintegers,implementtwo7xC0oLCoAVfunctions'RGaussGCD/':(::: )'RandGaussLCM(::: )which'Rcompute7xthegreatestcommondivisorandleastcommonmultipleoftwo7xGauianUUintegers,respGectively*.)6)7xUsingIC0oLCoAn,IprogramafactorizationalgorithmIGaussFactor? (::: )7xforUUelementsUUz72Z[i].7xHint:۲ProGceedasforintegers,searchingfordivisorsinthesetofall7xelementsUUa8+bisuchUUthata^2S+b^2Ȳdivides'(zp).,T utorialT5:SquarefreeP9artsofPolynomials궲In4thistutorialweshallexplorehowonecane ectivelycomputethesquare-freepartofaunivqariatepGolynomialovercertain elds.ItwillturnoutthatthiseseeminglyinnoGcentproblemisinfactintrinsicallyrelatedtothestructureofUUthebase eldUUK.a)%Let-KIbGe-a eldofcharacteristicp>0,-andlet':K~4!KIbGe-themap%de nedUUbyUU'(a)=a^pR.ThemapUU'iscalledtheF robQeniusTmap.)1)7xShowUUthatthemapUU'isUUaringhomomorphism.)2)7xShowUUthatUU'isbijectiveifK qis nite.)3)7xDeducethatifKοis nitetheneveryelementhasauniquep^th "ProGot."Ifa eldܵKhascharacteristic0orhascharacteristicpB>0and,inaddition,$hasthepropGertythateveryelementhasa$p^th ѲroGot,thenK;@iscalledapQerfectB eld.Inthesequel,weletKkDzbGeaperfect eldandfy2eK[x]anon-zeropGolynomial.W*eusetheconventiongcd(fV;0)=fsanddenotethederivqativeUUofUUfhbyf^0Ȳ.?b)%Let~char"(K) =p>0.~Showthat~f^0H= 0holds~ifandonlyif~fisofthe%formUUfڧ=g[ٟ^p PforUUsomeg"2K[x].\Ic)%SuppGosethatʵK=Fqj,whereqI=ٵp^e ande>0,isa nite eldof%characteristic>pF>0(see>T*utorial3),andletf(ո2FK[x]bGeapolynomial%suchothatof^0ԑ=0.ExplainhowonecancomputeapGolynomialogN2ɵK[x]%suchUUthatUUg[ٟ^pC=f.%Hint:XShowthateveryterminthesuppGortofXfisoftheformx^ pZand%proveUU(c^pre 0ncmsy51S)^pfj=cforUUallc2K.?d)%W*riteHaC0oLCoA"GfunctionHPRoot9(::: )HwhichtakesapGolynomialHf>2+FpR[x]%suchthatf^0=0andcomputesapGolynomialg"2FpR[x]suchthatfڧ=g[ٟ^p+.\Ie)%ShowUUthatifUUfhisirreducible,thenwehaveUUgcdUW(fV;f^0Ȳ)=1.%Hint:UUDistinguishthecasesUUchar(K)=0UUandchar(K)=p>0.& Wō>;384`1.F:oundations-ō>;f)%Letg"2K[x]bGeapolynomialsuchthatgcd(fV;g[ٲ)=1.Provetheformula %gcd4(fg[;(fg)^09)=gcd(fV;f^0Ȳ)gcd(g;g^0*).g)%LetwcharM(K)=0wandf=cQލ qs% qi=1;hp: i;ZiQbGewthefactorizationofwfinto%distinctYirreduciblefactors,whereYcȸ2K:nf0g.YShowthatgcd[(fV;f^0Ȳ)=%Qލ/#hs%/#hi=1>B_p: i*1;Zikand{deducethat{sqfreet(f){canbGecomputedbyusingthe%formula$sqfree9(f)=f=gcd(fV;f0Ȳ)?h)%FindUUanexamplewhichshowsthatg)isfalseifUUchar(K)=p>0.i)%Now)Llet)LKhbGea nite eldofcharacteristic)Lp>0.)LInPropGosition3.7.12%weGshallprovethatGsqfree]K(f)GcanbGecomputedusingthefollowingalgo-%rithm.)1)7xComputeUUs1C=gcd(fV;f^0Ȳ).UUIfs1=1,UUthenreturnUUf.)2)7xCheck(whetherwehave(s^0l1C=0.Inthiscase,useb)toconcludethat7xs1C=g[ٟ^p[for`somepGolynomial`g"2K[x].Compute`g]usingPRootu(::: ).7xThenUUreplaceUUfhbyfg);fe'gs1X= 8fK);fe+g@Lp1Y(,andcontinuewithstep1). S̍)3)7xComputej6si+1^3=gcd(siTL;s^0;Zi)fori=1;2;:::untils^0;Zi+1^3=0,j6i.e.until7xsi+1:issasp^th ~pGowersi+1na=g[ٟ^p oforssomegU2K[x].sThencalculateg7xagain,UUreplaceUUfhbyfg);fe'gs1,andcontinuewithstep1).%W*riteaC0oLCoA!HfunctionSqFree(::: )whichcheckswhetherthebase eld%is6QorFpkand6computesthesquarefreepartofagivenunivqariatepGoly-%nomial.,T utorialT6:Berlek\ramp'sAlgorithm궲Inthecaseofa nite eldK,weshallexploreaconcretealgorithmwhichfactorspGolynomialsinK[x].So,letpbGeaprimenumber,letebeapositiveinteger,q(letq(qQQ=xp^eKK,andletK(DbGethe eldwithqelements(seeT*utorial3).(If#youareunfamiliarwith nite elds,itisenoughtoconcentrateonthecaseUUq"=p,K~4=Z=(p).)"Our-goalistocomputethefactorizationofanon-constantmonicpGolyno-mialUUfڧ2K[x]ofUUdegreed=deg(f).a)%Prove#5thatthering#5R1=7K[x]=(f)is#5ad-dimensionalK-vectorspace%withUUbasisUUf1;MSxa;MSx^2 7;:::;MSx^d1Gg,wherexaƲistheresidueclassofxinRDz."InMwhatfollows,letMQef=(qij )bGeMthedޙd-matrixMoverKwhosei^th궲row1consistsofthecoGordinatesofԑ1x^q@L(i1)%?inthebasis1f1;MSxa;:::;MSx^d1GgofRDz.F*urthermore,(welet(g ׂ='w 0߲+_li+_l d1Ѳ&x^d1(with( 0|s;:::; d1 2KFDbGe(therepresentationUUoftheresidueclassofapGolynomialUUg"2K[x]inUUthisbasis.?b)%Showithat/ig[ٟ^q*=( 0|s;:::; d1&)? Q(1;MSxa;:::;MSx^d1G)^tr.iConcludethatthere%isa1-1correspGondencebetweenelementsB]ϵg 27RsuchthatB]ϵg[ٟ^q͸g }=70%andGvectorsG( 0|s;:::; d1&)2K^d suchGthatG( 0|s;:::; d1&)o(QId)=0,%whereUUIddenotesUUthed8dUUidentitymatrix.'ϠWō>;1.2UniqueTF:actorizationٮ39-ō>;\Ic)%F*oraanypGolynomialag7h2ۏK[x]satisfying+g[ٟ^q.%Ag ==0inRDz,aprovethatwe %have)1f;=(0Q 2K! jgcd0 l(fV;g!).)1(Hint:Findthefactorizationofx^q0x%andUUsubstituteUUg.forx.)?d)%Letffڧ=Qލ 8r% 8i=1Wصp: i;Zi n@bGefthefactorizationoff,where id>0fori=1;:::;r%andHp1|s;:::;pr %areHthedi erentirreduciblemonicfactorsoff.Prove%the:followingspGecialcaseoftheChinese2[RemainderTheorem.:The%canonicalUUmap-l":R߸U!K[x]=(p: 1l1 )8gK[x]=(p r፴r )%isvAanisomorphismofvAK[x]-algebras.(Hint:T*oshowsurjectivity*,let %g1|s;:::;grD!2K[x],-use-hi*˲=Q HHjg6=i!xpW j4=j oC,getanequation-Pލhr%hi=1_aiTLhi*˲=1, g$%andUUconsiderUU"(Pލ ;r% ;i=12giTLaihi).)%DeduceUUthatUU"inducesanisomorphismofK-vectorspaces-}ߣ':fWg2R߸jg[ٟqpn8g ۲=0g!Kr\Ie)%Conclude^fromd)thatthenumbGer^ofdistinctirreduciblefactorsof^fris %givenbyr=Ddim/K(ker \s(Qk Id)).W*riteaC0oLCoA"f!functionIsIrred)`(::: )%whichUUcheckswhetheragivenpGolynomialUUfڧ2K[x]isirreducible.%Hint:朲Y*oumayusetheC0oLCoA"/functionSyz(::: )tocomputethekernel%ofUUalinearmap.f)%ConsiderUUthefollowingsequenceofinstructions.)1)7xComputeUUthematrixUUQandUUthenumbGerUUrrde nedUUabove.7xLetDf(vi1P;:::;vid})V_j1irGgbeDaK-basisofkerZ(QؖId)and7xgid=vi1 /+̵vi2Px+h?+vid}x^d1>2K[x](Kfor1irG.(KW.l.o.g.wecan7xassumeUUthatUUgr4=1anddegx(giTL)>0for1i;404`1.F:oundations-ō>;%Hint:°T*oshow niteness,usethatthedeterminantofthematrix°of°the %map'abGoveisnon-zero,andthatthenumberofnon-constantdi erent%factorsinfi1gPfiiXisequaltothenumbGerofdi erententriesinthei^th%columnUUofUU.g)%ImplementBerlekqamp'sAlgorithmforKC̲=Z=(p).Thenapplyyour%function~Berlekamp5c(::: )~and~checkitagainstthebuilt-inroutinesof%C0oLCoAD+inUUthefollowingcases.)1)8xf1C=x^10098x^200;{1.3MonomialTIdealsandMonomialMoAdulesٮ41-ō>;1.3**MonomialIdealsandMonomialMo`dulesMathematicsN;424`1.F:oundations-ō>;\Ie)%Asubset˵B+Xiscalledasystem7ofgenerators %ofTifUUZ=f 8sj UP2G;s2Bqg.["Later6{thesymbGols6{andwill6{sometimesbeomitted.Obviously*,amonoidealXinXPisalsoac-monomoGdule.Infact,itisac-submonomoGduleof/the/c-monomoGdule,/justlikeforregularidealsandmoGdules.ThemostimpGortant exampleofamonomoduleforourpurposesisthesetoftermsT^nq~he1|s;:::;ermi-of-afreemoGduleRDz[x1|s;:::;xnq~]^rover-somepGolynomialring.ItisMxaMxT^nq~-monomoGdulegeneratedbyfe1|s;:::;ermg.SomemonomoGdulesrequirein niteUUsystemsofgenerators,asournextexampleshows.ExampleT1.3.2.c0 t,thesetofpGositiverationalnumbGers,isamonoidealinQ0B]whichisgeneratedbyf 133&fe~n ojn1g.ItiseasytoseethatthisUUmonoidealisnot nitelygenerated."Now@let@I=RnQbGe@thesetofirrationalnumbers.@AddinganelementofQ0òtoOanelementofOIyieldsOanelementofOI.Sinceconditions1)and2)ofDe nition1.3.1.caresatis ed,wehavehereanexampleofaQ0 t-mono-moGdule.cAgainonecanshowthatthismonomoduleisnot nitelygenerated.De nitionT1.3.3.is~LetUU(G;)bGeUUamonoidandUU(;)ac-monomoGdule.a)%W*e#saythatthecancellation-5la9wholdsin#c,if 1 3C= 2 3?implies% 1C= 2ȲforUUallUU 1|s; 2; 3C2c.?b)%W*eKsaythattheleft-cancellation/la9wholdsinK,if Cٵs1yy= s2%impliesUUs1C=s2ȲforUUall UP2c,s1|s;s2C2.\Ic)%W*esaythattherigh9t-cancellation1lawholdsin,if 1nysM= 2nys%impliesUU 1C= 2ȲforUUall 1|s; 22c,s2."IfEweconsideramonoidEasac-monomoGduleintheobviousway*,con-ditionsb)andc)bGothagreewitha)sothatthereisonlyonecancellationlawinʵc.InExample1.3.2,thecancellationlawholdsinQ0 t,andbGoththeleft-cancellationlawandtheright-cancellationlawholdinthemonomoGduleI.F*urthermore,-forevery-n)1,-thecancellationlawholdsinthemonoidoftermslT^n JintroGducedlinDe nition1.1.10,and,foreverylr51,bGoththeleft-cancellation lawandtheright-cancellationlawholdinthe T^nq~-monomoGduleT^nq~he1|s;:::;ermi.@ThefollowingconceptprovidesanimpGortant nitenesscon-ditionUUformonoids.PropQositionT1.3.4.qF;orAamonoidA(G;)theAfollowingc}'onditionsareequiv-alent.~oa)%Everymonoide}'alinvis nitelygenerated.Hb)%Every=asc}'endingchain=1døP294of=monoide}'alsin= iseventually%stationary.Hc)%Everynon-emptysetofmonoide}'alsinhasamaximalelementwith%r}'especttoinclusion."Ifthesec}'onditionsaresatis ed,themonoidviscalledNoQetherian.+*Wō>;{1.3MonomialTIdealsandMonomialMoAdulesٮ43-ō>;Pr}'oof.6Firstnweshowna)B)b).nSuppGosewehaveachainn1B2OWof monoidealsinandasequencen1C<n2<ֲsuchthatthereexistelements궵 id2ni+1nnyni forp allp i1.Thenweclaimthatthemonoidealgeneratedby궸f 1|s; 2;::: UOgisnot nitelygenerated.Itiscontainedintheunion[i1 iTL,butnotinoneofthemonoidealsصiTL.Nowassumethatitisgeneratedbya niteset.UUThensucha nitesethastobGecontainedinsomeUUiTL,acontradiction."Nowyweproveyb))c).LetySrbGeanon-emptysetofmonoidealsinc,andlet1C2S.If1Zkisnotmaximal,thereexistsamonoideal2C2Sqsuchthat1ds2|s.Continuinginthisway*,weobtainachain1ds2궲which uhastobGe nitebyb).ThenthelastelementofthechainisamaximalelementUUofUUS."T*oEshowtheremainingimplicationEc))a),EweletE/bGeEamonoideal.Theqsetofallmonoidealsinqgwhicharegeneratedby nitesubsetsofq궲containsYamaximalelement.Byconstruction,thiselementhastobGeYitself.X DPropQositionT1.3.5.qF;orn1,themonoid(N^nq~;+)isNo}'etherian.Pr}'oof.6W*euseinductiononn.WhennPE=1,everymonoidealisobviously of_theform_(a)with_a xed_aG2N._F*orn>1,_welet1TG2⓲bGe_anascendingg9chainofmonoidealsing9N^nq~.SuppGosethereareindicesn1C<n2<궲and[3elements[3wiԸ2{ni+1nsni fori1.[3Letv1=wm1C2fw1|s;w2;::: UOg궲bGeavectorwhose rstcomponentisminimal.Thenweletҵv2 q=wm2tL2fwm1 +1;wm1 +2;::: UOgSbGeSavectorwhose rstcomponentisminimalagain,etc.Inthiswayweconstructasequencev1|s;v2;:::CofvectorsofN^n `+whose rstUUcompGonentsformanon-decreasingsequence."F*oreallei 1,weletev^[ٷ0;ZihnowbGethevectorineN^n1whichconsistsofthelastnp1compGonentsofviTL.BytheinductionhypGothesis,thechainofmonoideals(v^[ٷ0l1|s)3(v^[ٷ0l1;v^[ٷ0l2)inN^n1CbGecomeseventuallystationary*.Thenalsothechain(v1|s)(v1;v2)c²ofmonoidealsinN^n cbGecomeseven-tually`stationary*,sincethe rstcompGonentsof`v1|s;v2;:::8form`anincreasingsequence.uW*earriveatacontradictiontotheconstructionofuw1|s;w2;:::UO,usinceweUUhadUUvid=wmi= 2o(w1|s;:::;wmi1b)(v1;:::;vi1 )UUforUUalli2.%+U"궲IntheremainingpartofthissectionweshallapplytheabGovetheorytothe!YmonoidoftermsinapGolynomialring.ThetranslationofthepreviouspropGosition7providesuswithanimportant nitenessconditionforidealsinpGolynomialUUrings.CorollaryT1.3.6.f(Dic9kson'sTLemma)L}'et/׵n1,/andlett1|s;t2;:::&b}'e/asequenceoftermsin/T^nq~.Thenthereexistsadnumb}'erdN>ٲ0suchthatforeveryi>Nthetermdtiisamultipleofoneofthetermst1|s;:::;tN,i.e.themonoide}'al(t1|s;t2;::: UO)j+T^n _jisgener}'atedby궸ft1|s;:::;tNg."InDp}'articular,foreveryringDR,theidealD(t1|s;t2;::: UO)CRDz[x1|s;:::;xnq~]Dis nitelygener}'ated.,;ݠWō>;444`1.F:oundations-ō>;Pr}'oof.6Themaplog:T^n q@!N^n givenbyx: 1l1 Dx^ n፴n7!²( 1|s;:::; nq~)is clearlymanisomorphismofmonoids.Themonoidealm(log (t1|s);logT(t2);::: UO)N^n궲isd nitelygeneratedbythepreviouspropGosition.ThusthereexistsanumbGer궵N3>0 such thatthismonoidealisgeneratedby flog (t1|s);:::;logT(tN)g.Con-sequently*,UUthemonoidealUU(t1|s;t2;::: UO)T^n ӲisUUgeneratedbyUUft1|s;:::;tNg. "궲As=weshallsee,idealsandmoGdulesgeneratedbytermshavemanyspGecialpropGerties.<W*ebeginourstudiesbygivingthemaspecialname.Let<ROʲbearing,UUletUUn1,letUUP*=RDz[x1|s;:::;xnq~]bGeUUapolynomialring,andletUUr51.De nitionT1.3.7.is~A: Pc-submoGdule:TM3Pc^r is:TcalledamonomialmoQdule,ifithasasystemofgeneratorsconsistingofelementsofT^nq~he1|s;:::;ermi.AmonomialUUsubmoGduleofUUPisalsocalledamonomialTidealofPc."MonomialIqidealscanbGereadilyvisualized,especiallywhentherearejusttwoUUorthreeindeterminates.RemarkT1.3.8._uHF*orrmonomialidealsrIkRDz[x1|s;x2],rwecanillustratethesetoftermsinIasfollows.Atermx^il1|sx1ɍjxݍ2ﶸ2sCT^2 9isrepresentedbythepGoint (i;j)2N^2|s.Then,foreachtermﴵx^il1|sx1ɍjxݍ2C2I,thequadrantﴸfx^kl1됵x^ll2Cjki;lxjg궲isg[containeding[I.F*orinstance,whenI=!(x^5l1|s;x^3l1x2;x1x^2l2;x^4l2)g[weg[obtainthefollowingUUpicture. t qϹfft@B33ffَXN9.9N9.N9. N9.sN9.FN9.DܞN9.rN9.N9.~N9.4N9.OʞN9.`N9. 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s[o25f1;:::;rGgLVsuchLVthatwehave%M3=ht1|se 1u;:::;tsF:e si.Hb)%Ther}'e8?aremonomialideals8?I1|s;:::;Ir^rеPsuchthatMOZisoftheform%MT͍3+33=li^r;Zi=1 tOIiTLeiTL.-QWō>;{1.3MonomialTIdealsandMonomialMoAdulesٮ45-ō>;Pr}'oof.6LetDBlT^nq~he1|s;:::;ermibGeDasystemofgeneratorsofDM.F*orevery numbGerEQiW2f1;:::;rgEQweEQde nethesetBi\=Wft2T^n Ȏjtei2BqgT^nq~.ByDickson'sLemma,themonomialidealsIi9=R(BiTL)have nitesystemsof7agenerators7aGi"?ֵBiTL.ObviouslythePc-moGduleMN|isthengeneratedby궵G1|se13[LD'[LGrmer jxT^nq~he1;:::;eri.'Thisproves'a)andtheclaim'M=궟Pލxr%xi=1-IiTLei_}in 1b).Thefactthatthissumisdirectfollowsfrom 1M3^r;Zi=1 tOPceiTL.X DǗ"궲Thel rstpartofthistheoremsaysinparticularthattheanalogueofPropGosition1.3.4.aholdsformonomialmodules.LetusnotethatthisimpliesthatUUalsotheanalogueofPropGosition1.3.4.bistrue.LCorollaryT1.3.10.lEveryasc}'endingchainofmonomialsubmodulesofPc^r Liseventuallystationary.Pr}'oof.6SuppGoseethereexistsastrictlyascendingchaineM1OܵM2ofmonomialHsubmoGdulesofHPc^r1.Sinceeachmoduleisgeneratedbyterms,wecannthen ndatermnti2@MiU2fiTLti궲with?pGolynomials?f1|s;:::;fs )2xPc,thenthetermtmustshowupinthesuppGortUUofoneoftheelementsUUf1|st1;:::;fsF:tsF:."T*oGshowb),weproveexistence rst.ByTheorem1.3.9.a,thereexistsa niteq(systemofgeneratorsofq(MCconsistingofterms.IfwedeleteinthissetallUtermswhicharepropGermultiplesofanotherelementofthatset,andifwealsoremoveallrepGetitionsofanelement,weobtainasystemofgeneratorsofUUMlpwhichUUcannotbGeshortenedanymore."T*oproveuniqueness,wesuppGosethattherearetwodi erentminimalmonomialsystemsofgeneratorsG1 mqandG2ofM.Bysymmetry*,wemayassumethatthereisatermt2G1&nG2|s.F*roma)weconcludethattisamultiplePofanelementPt^0Q2G2|s.Usinga)again,weseethatPt^09,andthereforet,isLamultipleofoneoftheelementsofLG1|s.SinceG1ȇisminimal,thatelementisnecessarilytitself,i.e.tandt^0bRaremultiplesofeachother.Thust=t^0Q2G2|s,aUUcontradiction.z.Wō>;464`1.F:oundations-ō>;%ExerciseA1.LetbAeacomm9utativegroup.Sho9wthatinitthereis %onlyTonemonoideal,namelyT^7itself.%Exercise.2.]Equiprthesetr=f0;1gwithrthe\natural"additionand%sho9wEPthatEP(A;+)isacomm9utativemonoidinwhichthecancellationlaw%doAesTnothold.%Exercisec3.W:econsidertheadditiv9emonoidQ0 (seeExample1.3.2).%W:eTletTabAeanon-negativ9erealnumbAerandTQa ~=fb2Q0 jbag. *?a)7xPro9vexthatifxa2Q,xthenQaj0isaprincipalmonoideal,i.e.a7xmonoidealTgeneratedb9yasingleelement.)b)7xPro9vexthatifxa=2 LQ,thenQa0isamonoidealwhic9hisnot nitely7xgenerated.*6c)7xIfa=2 QQ, ndanin niteincreasingsequenceofmonoidealsinQ>07xwhoseTunionisTQa S.)d)7xPro9veTthatthemonomoAduleTI=R8nQisTnot nitelygenerated.%ExerciseT4.jLetTususethenotationofExample1.3.2again. *?a)7xSho9wTthatTQ>0withtheusualmultiplicationisamonoid.)b)7xSho9wTthatthismonoidhasnonon-trivialmonoideal.*6c)7xSho9wTthatTI=R8nQisTaQ>0 -monomoAdule.%Exercise5.LetX(A;)beXaNoetherianmonoidinwhic9hthecancellation%la9wxholds.Assumingthattheonlyunitisx1,provethateverymonoideal%ThasTauniqueminimal(i.e.shortest)setofgenerators.%Exercise6.LetD(A;)beDamonoid,DaD nitelygeneratedmonoideal%inH,andletB/TbAeasystemofgeneratorsof.Pro9vethatcanbAe%generatedTb9ya nitesubsetofB.%ExerciseC7. |HLet n1andr1. Sho9wthatthesetofterms%T-=n7he1*;::: ;er,piTisTamonomoAduleo9verT-=n7.%Exercisei8.տLetBsWT-=n SbAesuc9hthatnoelementinBisdivisibleby%anotherTelemen9tinTBr.ProvethatTB is nite.%Exerciseo9.mpLet(A;)beamonoid,andletbeaH-monomodule.%W:esa9ythatisaNoKetherianH-monomoAdule+ifev9eryascendingchain%ofTH-submonomoAdules1m2qofTisTev9entuallystationary:. *?a)7xF:or%BsubmonomoAdulesof%B2,form9ulateandproveananalogueofPropAo-7xsitionT1.3.4.)b)7xLet%nbAe%aNoetherianmonoid,andlet% bea nitelygenerated7xH-monomoAdule.TThensho9wthatTisNoetherian.*6c)7xConcludeTthattheTT-=n7-monomoAduleT-=n7he1*;::: ;er,piisTNoetherian./Wō>;{1.3MonomialTIdealsandMonomialMoAdulesٮ47-ō>;T utorialT7:Cogenerators궲Let(G;)bGeamonoid,letbeamonoidealinc,letu=cnbGethe complementrofrinc,andletC幸..W*esaythatCJcogeneratesif궵=f UP2*j 8 8^0#2C qforUUsome,h 8^02cg.s󍍍~jq(ff~ǝ33ff}0EN9.̞N9.ٞN9.}N9.N9.NN9. 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(Ancien9tTLatinProverb)"궲LetusforamomentgobacktoSection1.1wherewediscussedthenotion of pGolynomialringsinoneandseveralindeterminates.Aunivqariatepolyno-mialڌwithcoGecientsinaringڌRSisanexpressionofthetypGeڌf(x)=PriTLx^iTL.Onequestionis:inhowmanydi erentwayscanwewritef(x)?W*emightagreeHthatthecoGecientsshouldbewrittenbeforethecorrespondingpowerproGductandalsodecidetobe\nice"andavoidthe+signbGeforethe rstco-ecient,butstillwehavetofacethecommutativepropGertyofthesum,whichimpliesthatforinstance1Y+2x3x^2 &canalsobGewrittenas1Y3x^2Ւ+2x.This~maynotbGearelevqantquestionfor\pure"mathematicians,butitisfun-damental*theonedescribGedby>xR<1.>ThisistheendofthestoryforunivqariatepGolynomialsRandalsoformultivqariateones,ifarecursiverepresentationisused."ButwehavealreadyseenthatotherpropGertiesofpolynomialsallowustogetridoftheparenthesesandexpressthemassumsofcoGecientstimeselementsuDinuDT^nq~.SothequestionnowishowtoorderuDT^nq~.F*orthesamereasonsashbGeforeweneedcompatibilitywithitsmonoidstructure.Lethf(x1|s;x2;x3)=궵x1|sx3}+3x^2l2|s;9~shouldwewriteitas9~x1x3}+3x^2l2orratheras9~x^2l2+3x1|sx3|s?Thereisnoobviousanswertothisquestion,andthepurpGoseofthissectionistoshedsomeUUlightonit.2uWō>;504`1.F:oundations-ō>;"궲Inparticular,weshallstudytotalorderingsonT^n 3andonT^nq~he1|s;:::;ermi. If[Ttheyhave[TacertainadditionalpropGerty*,theyarecalledtermorderings.This\fundamentalpropGertyofmoGduletermorderings"isthekeytoshowing nitenessUUformostalgorithmsweshallencounterlater."AnattempttoclassifyallpGossibletermorderingsonT^nq~,atleastinsomeeasycases,ismadeinT*utorial10.AlthoughweavoidtreatingthegeneralRclassi cationoftermorderings,wedoshowthatsomeorderingscanbGede nedbymatricesviascalarproducts(seeProposition1.4.12),andthatallUUthemostimpGortantmonoidorderingsareofthattypGe."In4thefollowing,let4(G;)bGe4amonoid.RecallthatforusthisalwaysmeansUUthatUUiscommutative.De nitionT1.4.1.is~Arelation; onʲis;asubsetof;T=c.Ifapair( 1|s; 2)is^inthatsubset,weshallwrite^ 1 C f 2|s.Arelation)7on0iscalledcompleteeifanytwoelementse 1|s; 2 2{ɇarecomparable,i.e.ifwehave궵 1C) 2ȲorUU 2 1|s."Azcompleterelation }on3iscalledamonoidorderingifthefollowingconditionsUUaresatis edforallUU 1|s; 2; 3C2c.a)& 1C) 1(re exivity)?b)& 1C) 2ȲandUU 2 1ȲimplyUU 1= 2P7u(antisymmetry)\Ic)& 1C) 2ȲandUU 2 3ȲimplyUU 1 3V(transitivity)?d)& 1C) 2ȲimpliesUU 1S8 3 2S8 3궲If,UUinaddition,wehave\Ie)& UP)1 forUUallUU 2궲thenUU.isUUcalledatermTorderingonc."Ifӵjisarelationonc,andif 1|s; 2C2rbaresuchthat 1C) 2|s,wealsowrite6; 2^ 1|s.6;F*urthermore,ifadditionally 1^6== 2|s,wewrite 1^> 2궲or` 2 ߵ<  1|s.`Ifthecancellationlawholdsinc,condition1.4.1.dcanbGereversedUUasfollows.RemarkT1.4.2._uHLetUU.bGeUUamonoidorderingonc.a)%SuppGose?#thatthecancellationlawholdsin?#c,andlet 1|s; 2; 3C2c.?#Then%anTinequalityT 1и7] 3C) 2 3impliesT 1C) 2|s.TThisfollowsfromthe%observqationthat 2C>) 1fimplies 2H 3) 1 3fbyDe nition1.4.1.d,%andUUequalityisexcludedbythecancellationlaw.?b)%IfSh6=nٸf1nøgandSthecancellationlawholdsinSc,then&isin nite.%Namely*,SletS 6=n1 KbGeanelementofc.Nowletusconsidertheset%Sֲ=If 8^i͸ji2Ng.uW*ehaveueitheru1 l >_ߵ or >1nò.uInthe rstcase%1 3 > & R> 8^2>showsLthatLSٲisin nite.Inthesecondcasewe%argueUUanalogously*.\Ic)%Byinductionwecanshowthat,forany c>2tandanyn>0,the%conditionUU UP)1 isUUequivqalentto 8^nθ)1nò.30Wō>;z1.4T:ermTOrderingsٮ51-ō>;"궲Underzxtheisomorphismofmonoidszxlog:T^n !#!N^nq~,zxmonoidorderings (resp. termorderings)on T^n VcorrespGond1-1tomonoidorderings(resp.termorderings)onN^nq~.NowweintroGducesomeofthemostimportanttermor-deringsUUonUUT^nq~.De nitionT1.4.3.is~F*orеt1|s;t22iT^n (Nwesayt1/Cscmtt8Lext2 3Cifandonlyifthe rstnon-zerocompGonentoflog}<(t1|s) \log(t2)ispGositiveort1 TA=εt2|s.Itiseasytorcheckthatthisde nesatermorderingonrT^n |itiscalledthelexico-graphicTtermorderingUUandisdenotedbyUULexL.ExampleT1.4.4.cLexMx2>Lex >Lexxnq~.t`F*orinstance,whent`n~=3,t`wehave궵x1|sx^2l2:>Lex еx^3l2x^4l3|s,MsinceM(1;2;0)Y(0;3;4)d=(1;1;4)hasMapGositive rstcompGonent.nSimilarly*,wehavenx^3l1|sx^2l2x^4l3_>Lexx^3l1x^2l2x3|s,nsincethe rstnon-zerocompGonent5of5(3;2;4)Ң(3;2;1)=(0;0;3)is5pGositive.Alsox1|sx3C>Lexo!x^2l2|s,andweseehowtouseLex?toorderthepGolynomialmentionedatthebGeginningofUUthesection."F*or n=26, ifonereplacesx26 byA,x25byBq, etc.,andonedecidestowrite[8\smallest rst",thenthelexicographicorderingonthetermsbGecomessimilartotheusualorderingonwordsinadictionary*.WesaysimilarandnotequalbGecausethereisafundamentaldi erencebetweenourwordsandthewordsH[inadictionary*.Ourwords(orterms)arecommutative,soinourlexiconthetwowordsap}'elZandpeapbarethesame.AlthoughthisbGookisentirelyaboutcommutativethings,wemustadmitthatnon-commutativedictionarieshavecertainUUadvqantages.De nitionT1.4.5.is~F*or;"twoterms;"t1|s;t2C2T^n wesay;"t1CDegLex*t2ifwehavedeg#E(t1|s)>degt(t2),rorifwehaverdeg(t1|s)=degt(t2)randt1sXLext2|s.rItiseasyto6checkthatthis,toGo,de nesatermorderingon6T^n |itiscalledthedegree-lexicographicTtermorderingUUandisdenotedbyUUDegLex$C.ExampleT1.4.6.cDegLex*(>DegLex*xn holdsagain.rF*orinstance,whenrĵnMx=3,rwehaverĵx1|sx^2l2x^3l3 >DegLexx^2l1x^2l2|s,rsincedeg#E(x1|sx^2l2x^3l3)X=6>4=deg(x^2l1|sx^2l2),vandwehavevx^2l1|sx^2l2x^2l3 %˵>DegLexyjx1x^2l2x^3l3|s,since(deg(x^2l1|sx^2l2x^2l3)=6=deg(x1|sx^2l2x^3l3)(and(2;2;2)h(1;2;3)=(1;0;1)(hasaUUpGositive rstcomponent.De nitionT1.4.7.is~F*orѵt1|s;t2 }2T^n POwesayt1 }DegRevLex(t2 [Difwehavedeg#E(t1|s)q>deg(t2),v&orifwehavev°(t1|s)q=deg(t2)v&andv&thelastnon-zerocompGonent>of>log(t1|s)n&logX(t2)>is>negative,orift1ȹ=LFt2|s.Itiseasytocheckthatgthisde nesatermorderingongT^n q|itiscalledthedegree-rev9erse-lexicographicTtermorderingUUandisdenotedbyUUDegRevLex4:.ExampleT1.4.8.cDegRevLex'90ĵ>DegRevLex'xnq~."F*orinstance,whenn=3,"wehave궵x^4l1|sx^7l2x3 ˵>DegRevLex'sx^4l1x^2l2x^3l3|s,sincedeg*@(x^4l1x^7l2x3)X=12>9=deg(x^4l1|sx^2l2x^3l3),andwehavex1|sx^5l2x^2l3Ѭ>DegRevLex%MTx^4l1x2x^3l3|s,sincebGothtermshavedegree8and4Wō>;524`1.F:oundations-ō>;궲(1;5;2)⠸(4;1;3)=(3;4;1)*5has*5anegativelastcompGonent.Similarly*,we have6x^3l1|sx^3l2x^2l3<DegRevLex% Tx^4l1x^2l2x^2l3|s,6sincebGothtermshave6degree8andthelastnon-zeroUUcompGonentofUU(3;3;2)8(4;2;2)=(1;1;0)isUUpGositive."Ifwedropthe rstconditioninthede nitionofDegRevLex4[,i.e.ifwelet궵t1}RevLexbt2 Kifthelastnon-zerocompGonentoflog(t1|s)logt(t2)isnegativeorift1 z=t2|s,weobtainamonoidorderingonT^nq~,calledtherev9erse-lexicographic%ordering,ywhichisnotatermordering(seeExercise3).De nitionT1.4.9.is~ARmonoidRorderingRsonT^n iscalleddegree0compati-bleUUifUUt1C)t2Ȳfort1|s;t22T^n Ӳimpliesdegx(t1|s)deg(t2)."F*or_instance,_DegLex(> and_DegRevLex7aredegreecompatibletermorder-ings.De nitionT1.4.10.o3|LetY1j! 1+d+ j6,orif 1ٚ+d+ jXv=!ʵ 1+d+ jland궵t1 DegRevLex%t2|s.ʣItiseasytocheckʣthatthisde nesatermorderingonʣT^n궲|,itiscalledaneliminationorderingfor,õLand,isdenotedby,Elim(L)'3."TheIorderingsIElim(L),+aremembGersofalargerclassofeliminationor-deringslIdescribGedinSection3.4.Againtheindeterminatesareorderedby궵x1C>:Elim(L))*>:Elim(L))xnq~."LoGoking1attheseexamples,wenoticethattheyshareacommonprop-erty:thecomparisonoftwotermsisachievedbycomparingtheirlogarithms.Indeed,sincethemaplogF:T^n8!N^n isanisomorphismofmonoids,onecanuse1termsandtheirlogarithmsinterchangeably*.1ThusintheabGoveexamplestheDegRevLex%?nt2 ifandonlyifthe rstnon-zerocompGonentof( 1E+_ҵ:::jK+_ҵ nq~; n;:::; 2|s)ispGositive.W*eseethatthecomponentsofthisvectorarelinearfunctionsintheOcoGordinatesofOlog:(t1|s)-log(t2).OThisleadsustointroGducethefollowingconstruction.De nitionT1.4.11.o3|LetLv1|s;:::;vn .2MZ^n JʲbGeLlinearlyindependentvectors,andletVbGethenon-singularmatrixwhosei^th mrowisvi fori=1;:::;n.F*or궵t1|s;t2C2T^nB,QwesayQt1C:Ord ( :(V) t2[ift1=t2[orQifthe rstnon-zerocoGordinateof[thevector[V ò(log (t1|s)logo(t2))[is[pGositive,wheredenotestheusualmatrix-by-vectorproGductand(log (t1|s)logR(t2))hastobGeconsideredasacolumn̼vector.Itiseasytocheckthatthisde nesamonoidordering̼Ord(V8)onUUT^nq~.UUW*ecallittheorderingTrepresen9tedbyUUV8.5ɠWō>;z1.4T:ermTOrderingsٮ53-ō>;PropQositionT1.4.12.wL}'et*Vbe*thematrixwhoser}'owsarelinearlyindepen- dentve}'ctorsv1|s;:::;vn82Z^nq~.ThenOrd@(V8)isatermor}'deringifandonlyifthe rstnon-zer}'oelementineachcolumnofVispositive.ePr}'oof.6It8^isclearthatamonoidordering8^7onT^n ܲis8^atermorderingifandonlyifxit>71fori =1;:::;n.LetaiZbGethe rstnon-zeroelementofthe궵i^th columnofV8.ThenV(D(log (xiTL)log(1))=(0;:::;0;ai;::: UO)^tr showsthat궵xid>:Ord ( :(V) 1UUisUUequivqalenttoaid>0.="궲F*or %example,itiseasytoseethat %LexAistheorderingrepresentedbytheidentitymatrix,andfromtheabGovedescriptionofDegRevLex7!=itfollowsthatitUUisrepresentedbythematrix),rkȵV=T0BBBfi@1&19:::S11i10&09:::S10fQ10&09:::ON11i0 1:::#j:::9:::Om:::f:::0"i1<0Om:::i0Tt1tCtCtCfitA"궲Also fortheothermonoidorderingsintroGducedabove itispossibletoseethatPtheyarerepresentedbysomematrix(seeExercise6).Thereisacompleteclassi cation7Lofmonoidorderingson7LT^nq~.Itsaysthattheyessentiallyareallof3VtypGe3VOrdM(V8),where3VVl:isamatrixwithentriesin3VR(see3VExercise7).F*orcomputationalxpurpGoses,monoidorderingsrepresentedbyintegralmatricesasUUabGovearegoGodenough."OurnexttwopropGositionsdealwiththequestionhowtermorderingsbGehaveWunderrestrictionsandextensionsofthemonoidsonwhichtheyarede ned.ePropQositionT1.4.13.wL}'etf/?befamonoidor}'deringonT^nq~,andletT䍴n1iXb}'ethemonoidoftermsintheindeterminatesx1|s;:::;xi1 ;xi+1 tO;:::;xnq~.~oa)%Ther}'estrictioni3oftoT䍴n1iAisamonoidor}'dering.Hb)%Ifisatermor}'deringthenalsoi3isatermor}'dering.Hc)%Supp}'osew$thatw$isrepresentedbyamatrixw$V8.Thenipisr}'epresentedw$by%thematrixVi?whichisobtaine}'dfromVby rstdeletingthei^th c}'olumn%andthenthe rstr}'owwhichislinearlydependentonthoseaboveit.Pr}'oof.6The rstassertionisclearsinceT䍴n1icanbGeviewedasasubmonoidofT^nq~.Thesecondonefollowsimmediatelyfromthede nitionofatermordering.$T*oprove$c),weobservethatifwedisregardthe$i^th /Eindeterminate,wemustdeletethe޵i^th columnfromV8.ThenweareleftwithamatrixofshapGe#ܵn(n1)and#rankn1.#W*edeletethe rstrowwhichdepGendsonthoseabGoveit,bGecauseifforavectorallthepreviousscalarproGductsvqanish,thenalsothescalarproGductwiththedependentrowvqanishes.ThuswegetamatrixVis ofshapGe(n&1)(n1)andrankn1.ItrepresentsiTL,bGecause forall t1|s;t2vӸ2`T䍴n1ithevectorViɲ(log (t1|s)logu(t2)) agrees with궵VǸX(log (t1|s)logC(t2)),ZexceptthatwehavetoregardZt1 Ͳandt2asZelementsofUUT^n ӲandUUtoremoveUUtheentrycorrespGondingtothedeletedrow.'BG6(lWō>;544`1.F:oundations-ō>;PropQositionT1.4.14.wEvery}monoidor}'dering}VonN^n has}auniqueexten- siontoamonoidor}'dering[ٟ^0onZ^nq~.Pr}'oof.6F*orsvx2jZ^nq~,sthereexistvectorsv1|s;v22jN^n (suchsthatvx=jv1zIv2|s.W*esayv@L0 Iв0ifandonlyifv1:]v2|s.T*oseethatthisiswell-de ned,wetaketworepresentationsv =(1v1*_v2=v^[ٷ0l1_v^[ٷ0l2 withv1|s;v2;v^[ٷ0l1;v^[ٷ0l22(1N^n andnoteCthatCv1:v2 Aisequivqalenttov1+~v^[ٷ0l2:v2+v^[ٷ0l2 AbyCRemark1.4.2.a.Thisninturnisequivqalenttonv^[ٷ0l1b+Iv2n0TSv2+v^[ٷ0l2|s,nandthereforetonv^[ٷ0l1n0TSv^[ٷ0l2.If wenowde ne v"@L0 !w(UX)0vFmw @L00forv[;w 2Z^nq~, itiseasytocheckthatUUAxiomsa){d)ofDe nition1.4.1aresatis ed."The7euniquenessof7e[ٟ^0awfollowsfromtheobservqationthat,forv[;v^0i2?Z^n궲such9that9v"=v1Nv2\andv[ٟ^0*=v^[ٷ0l1Nv^[ٷ0l2\withv1|s;v2;v^[ٷ0l1;v^[ٷ0l2C2N^nq~,9thecondition궵v"@L0 !v[ٟ^0gisUUequivqalenttoUUv1S+8v^[ٷ0l2C)v^[ٷ0l1+v2|s."궲In֚viewofthispropGosition,andbyextending֚Lexm+to֚Z^nq~,wecanrephraseDe nitiontH1.4.11asfollows:tHt1w:Ord ( :(V)Tt2 q˸(UX)XVfM(log (t1|s)log8.(t2))Lex0forUUt1|s;t2C2T^nq~."TherpropGositionalsoimpliesthatstudyingmonoidorderingsonrT^n 4isequivqalentjtostudyingmonoidorderingsonjZ^nq~.Inparticular,weshallusethe [samesymbGolforamonoidorderingon [T^nq~,itstranslationtoN^nq~,anditsunique(Hextensionto(HZ^nq~.Inparticular,wemayapplythisnotationalconven-tionMandsaythatforamonoidorderingM,&onT^n A˲andt1|s;t1}2 T^nweMhave궵t1C)t2 (UX)0logx(t1|s)8log#(t2))0."The nalpartofthissectiontreatstheextensionofthetheoryofmonoidorderings[toorderingsonmonomialmoGdules.Moreprecisely*,for[n;r51,wewantDtode nesuitableorderingsonthesetoftermsDT^nq~he1|s;:::;ermiintroGducedinUUDe nition1.1.10.De nitionT1.4.15.o3|Let(G;)bGeamonoidand(;)ac-monomoGdule.Acomplete=-relation=-onкiscalledamoQdule'orderingifforalls1|s;s2;s3C2궲andUUallUU UP2wehavea)&s1C)s1(re exivity)?b)&s1C)s2ȲandUUs2s1ȲimplyUUs1=s2V}"(antisymmetry)\Ic)&s1C)s2ȲandUUs2s3ȲimplyUUs1s3\8(transitivity)?d)&s1C)s2ȲimpliesUU 8s1 8s2궲If,UUinaddition,wehave\Ie)& 8s)sUUforUUalls2andallUU UP2궲thenUU.isUUcalledamoQduleTtermorderingUUon."F*or%us,themostimpGortantcasewillbethecase%=JT^n Handݨ=T^nq~he1|s;:::;ermi.IfrGβ=1,thenmoGduleorderingsaremonoidorderingsonT^n ST=T^nq~he1|siasintroGducedinDe nition1.4.1,andmoduletermorderingsarenothingbuttermorderings.W*ealsonotethatitiseasytoseethatinthiscaseWconditione)isequivqalenttoWteid)eiforallt2T^n 9ղandWalli=1;:::;rG.7;\Wō>;z1.4T:ermTOrderingsٮ55-ō>;궲The`mostimpGortantmoduleorderingsforourpurposesareconstructedas follows.ExampleT1.4.16.h`LetUUTo*bGeUUatermorderingonUUT^nq~.a)%F*orVt1|seiTL;t2ejĸ2T^nq~he1;:::;ermiVsuchVthatt1|s;t2C2T^n %Բandi;jY2f1;:::;rGg,%weUUlet5~t1|seidToPos't2ej޸(UX)2t1C>To t2|tor!g!(t1=t2ȲandCij)%InuthiswayuweobtainamoGduletermorderingToPosonuT^nq~he1|s;:::;ermi. %TheCintuitivemeaningofToPosisthatone rstcomparesthetwopGower%proGductszusingToandthenbreakstiesbylookingattheirpositionsin%theUUvector.?b)%F*orVt1|seiTL;t2ejĸ2T^nq~he1;:::;ermiVsuchVthatt1|s;t2C2T^n %Բandi;jY2f1;:::;rGg,%weUUlet9t1|seidPosTo't2ej޸(UX)2i u oθᖵsforsome궵 UP2{andus2,uthens>) KX s> 8^2˸ s>ʲisuanin nitechainwhichisUUnoteventuallyUUstationary*.PE8QWō>;564`1.F:oundations-ō>;TheoremT1.4.19.jUF(F undamen9talTPropQertyofT ermOrderings) oF;orƹamo}'duleorderingƹ"onT^nq~he1|s;:::;ermi,ƹthefollowingconditionsaree}'quivalent.~oa)%Ther}'elationisamoduletermordering.Hb)%Ther}'elationisawell-ordering.Pr}'oof.6In'viewofthepreviouspropGositionandthefactthattheleft-cancellation~lawholdsin~T^nq~he1|s;:::;ermi,itsucestoprove~a))b).~W*esuppGoseL%thereisachainL%t1|se 1  t2e 2 ӲinL%T^nq~he1;:::;ermiwhichL%isnoteventuallystationary*,wheret1|s;t2;:::2RT^n and 1; 2;:::2Rf1;:::;rGg.F*orpsomepi2f1;:::;rGg,wethenhaveasubGchainpt1 eiW9 et2ei e궲such/Ethat/E12Q1ĵ<2< =which/Eisnoteventuallystationary*.ByDick-son's;Lemma1.3.6,themonoideal;(t1 ;t2;::: UO);is;generatedby nitelymanytermsrt1 ;:::;tmNeforrsomeN#c> H0.SincetKisamoGduletermordering,itfollowsQthatforeachQjY>NithereexistsanumbGerQk2f1;:::;NgsuchQthat궵tjߵeid)tkeiTL,UUacontradiction.3j%Exercise1. 4%Pro9vel,thattherelationsl,Lex,l,DegLex"Ű,l,DegRevLex0r,and %Elim(L)HonTT-=n 3areTtermorderings.%Exercise82.GF:or8eac9hofthetermorderings8Lex:,8DegLex g,and8DegRevLex.~,%writeTdo9wnthe20smallesttermsofTT-=3?inincreasingorder.%Exercise73.hDe neEarelationERevLex$onET-=n cb9yt1 2Cscmtt8RevLex t2p&ifEthelast%non-zerocompAonen9toflog:(t1*)ologb^(t2)isnegativ9e,orift1.=t2*.Sho9w%thatRevLex&&isamonoidorderingonT-=n whic9hisnotatermordering.%Ho9wTaretheindeterminatesorderedbyTRevLex!n?%Exercise>4.oGohbac9ktoExample1.4.4,replacehx1byA,x2byBr,hetc.,%and#decidetowrite\biggest rst".Whatisthemonoidorderingon#T-=n%similarTtotheusualorderingonw9ordsinadictionary?%ExerciseS5. Letm7V{obAem7amatrixwhosero9wsarelinearlyindependen9tvec-%torsOv1*;::: ;vn2Z-=n7.OPro9vethattherelationOOrd(V8)Oisamonoidordering%onTT-=n7.%Exercise6.6F:orBeac9hofthetermorderingsBLex,BDegLex">,BDegRevLex0k,%andwElim(L)#y),wgiv9eanon-singularmatrixwVsuchthattheyarerepresented%b9yTV8.%Exercise7.$Let~u=(1;qp ?qaHr2 )and~let;(bAetherelationonT-=2%de nedb9y%t1mt2?ifTandonlyifTu8(log (t1*)log(t2))0Tfort1;t2m2T-=2*. *?a)7xSho9wTthatTgisatermorderingonT-=2*.)b)7xSho9wTthatTgcannotbAerepresentedbyanymatrixofintegers.7xHint:Pro9vethat,foranytermordering representedbyamatrixof7xin9tegers,TthereexisttermsTt;t-=0C2T-=2?suchthatTt>t-=0andt<At-=0.9bWō>;z1.4T:ermTOrderingsٮ57-ō>;%ExerciseX 8.ExPro9ve~xthatev9erymonoidordering~x"onZ-=n has~xaunique %extensionTtoamonoidorderingTR-=0onQ-=n7.%Hint:Av9ectorvp2Q-=n?canbAerepresentedintheformvp=P 1aH0wq s pwithq>0 %andYp2Z-=n7.YThende nevUffh7G0 0ifandonlyifpw0.T:oseethatthis%isEw9ell-de ned,taketworepresentationsEv5[=P 1aH0wq "T.p=P l1aHX-0q7G0 p-=0withqR;q-=0">0 `%andTp;p-=0C2Z-=n 3andTpro9vep0#(uV)qR-=0qp0.2%Exercise9. Sho9w!thattherelationsToPosandPosTode nedinExam-%pleT1.4.16aremoAduletermorderings.%Exerciseo10.qLetKgbAea eld,letPک=K[x1*;::: ;xn7],andlete1*;::: ;er%bAe8newindeterminates.Considerthe8PH-linearmap':PH-=r 3!PH[e1*;::: ;er,p]%de nedTb9yT'((f1*;::: ;fr,p))=f1e18+8+8fr,per,p. *?a)7xSho9wTthatT'isaninjectiv9ehomomorphismofPH-moAdules.%Using8',8w9eidentify8PH-=rwiththecorrespAondingsubmoduleof8PH[e1*;::: ;er,p]. %Let#bAeamonoidorderingonT-=n+r=T(x1*;::: ;xn7;e1;::: ;er,p),letbAe%the`monoidorderingon`T-=n 6obtainedb9yrestrictionof#(seePropAosi-%tionc1.4.13),andletcO bAetheorderinginducedb9y#onT-=n7he1*;::: ;er,pi%viaT'. )b)7xPro9veTthatTgismoAduleorderingonT-=n7he1*;::: ;er,pi.*6c)7xPro9veTthatTgiscompatiblewith.%Exercise11. Let:bAeamoduleorderingonT-=n7he1*;::: ;er,pi.View%T-=n7he1*;::: ;er,piIasIadisjoin9tunionofrcompAonen9ts,eachofwhichisa =n%cop9yTofTT-=n7,anddenotetherestrictionofgtothei-=th compAonen9tbyTi,r. *?a)7xPro9veTthatTiAisamonoidorderingforev9eryi=1;::: ;rA.)b)7xLetbAeamonoidorderingonT-=nsuc9hthatiscompatiblewith.7xPro9veTthatTi8=Sforev9eryi=1;::: ;rA.8T utorialT9:MonoidOrderingsRepresen9tedbyMatrices*궲Let(v1|s;:::;vnq~),(v^[ٷ0l1;:::;v^[ٷ0፴nq~)bGetwon-tuplesoflinearlyindepGendentvectors in +Q^nq~, +andletV9;V8^0 C2&Mat'bn(Q)bGethematriceshavingthosevectorsasrows.a)%ExtendY)De nition1.4.11toorderingsrepresentedbyrationalmatrices%likeUUV2MatTnFҲ(Q).?b)%SuppGosethereexistsalowertriangularmatrixWmP2 Matn{(Q)whose%entries0suchUUthatv^[ٷ0l1C=v1|s.:tWō>;584`1.F:oundations-ō>;g)%Nowweassumethat̵v^[ٷ0l16= 2v1 ?forall 2>0.W*riteaC0oLCoA#Tprogram %TODifferenced{(::: )whichcomputestwotermst1|s;t2 a2T^n s"suchthat%t1C>:Ord ( :(V) t2ȲandUUt1<:Ord ( :(V0mW)_t2|s.?h)%W*ritecaC0oLCoA#KprogramcCheckEqualityI(::: )cwhichchecksforagiven%numbGert׵d>0iftthetermorderingsrepresentedbyt׵VandV8^0 {agreetfor%allUUtermsofdegreeUUdinT^nq~.i)%(This>p}'artisamuchmoreelaborateproject.)Findcriteriawhichcharac-%terizeUUwhenUUV9andV8^0 \rrepresentthesamemonoidordering.,T utorialT10:Classi cationofTermOrderings궲Iny thistutorialwewanttogetagoGody understandingofallpGossibletermorderingsUUonUUT^n Ӳforn3.a)%ProveUUthatonUUT^1Ȳthereisonlyonetermordering,namelyDegL.?b)%ShowEthatonET^2$therearepreciselytwodegreecompatibletermorderings%͠;wwhichUUarecharacterizedbyUUx1C>)x2Ȳandx2>Mx1|s.\Ic)%ClassifyallpGossibletermorderingsonT^2ywhichsatisfyx1C>)x2|s.T*odo%thatUUusethefollowingscheme.)1)7xProvethatthereexistsexactlyonetermorderingNonT^2osuchthat7xx1C>)x^il2ȲforUUallUUi2.)2)7xSuppGosepFthatatermorderingpFonT^2satis esx1pr<Vx^Nl2 forpFsome7xN32,SandletSµq"=infƸf33j33);feq0i*2Q+ jjx^il1C<)x1ɍjxݍ2|sg.ProveSthatS1q"N.)3)7xF*ollowinga(2),suppGosethata(qN2uRlnQ.a(Showthatthereexistsex-7xactly:onetermordering:}withthosepropGertiesandthatitsatis es7xx:i1l1;lx:i2l2 )x1ɍj1xݍ1?x1ɍj2xݍ2 ifUUandonlyifUUi1|sq+8i2Cj1q+8j2|s.)4)7xF*ollowingZ2),suppGosethatZqi2 Qnf1g.ZShowthatthereexistex-7xactlytwotermorderingswiththosepropGerties.Provethattheyare7xrepresentedUUbymatricesbyexhibitingthematrices.)5)7xFinally*,if⧵qx=1in2),showthatthereexistsexactlyonetermor-7xderingUUwiththosepropGertiesandthatitisrepresentedbyamatrix.?d)%F*oralltermsofdegreeS2inT^3|s, ndallorderingsinducedbydegree%compatiblestermorderings.Y*oumayassumethattheindeterminatesare%numbGeredUUinsuchawaythatUUx1C>)x2>x3|s.\Ie)%RepGeatUUthepreviouspartforalltermsofdegreeUU3inT^3|s.f)%Provethattherearein nitelymanydi erentdegreecompatibleterm%orderingsUUonUUT^3|s.g)%W*riteYaC0oLCoA#!programYTermOrderListI>2(::: )YwhichtakestwonumbGers%i;dXC>0,ode nesadegreecompatibletermorderingoTO,ii-$onoT^3 (whichis%di erentforeachi,andreturnsthelistofalltermsofdegreedordered%accordingUUtoUUTOOi).;Wō>;1.5LeadingTT:ermsٮ59-ō>;1.5**LeadingTermsATheNitisautomaticallyconverted>to>x1|sx3{x^2l2+x1|s.>ThisdoGesnotviolate2thecommutativity2law,ratheritconveystheideathattheequality궵x1|sx3E"x^2l2+x10=/x^2l2+x1|sx3+x1 ,should-bGeinterpretedinthefollowingway*.ThepGolynomial͵f(x1|s;x2;x3),writtencorrectlywiththesequenceofsymbGolsx1|sx3ոNbx^2l2+x1|s,isequaltothepGolynomialx^2l2ղ+x1|sx3+x1|s,bGecause궸x^2l2s+x1|sx3+x1 Byisautomaticallyconvertedtox1|sx3sx^2l2+x1|s,andthisistheUUsamesequenceofsymbGolsasbefore."The1Ohierarchycreatedamongthetermsin1OSuppj7(f)1Obythemonoidor-deringimpliesthatx1|sx3 bGecomes\bigger"and\moreimportant"thantheother=terms.ShoulditbGecalledthe\leader",the\initial",orthe\head"?W*ecallittheleadingtermoff(x1|s;x2;x3).OfcourseallofthiscanandwillbGeUUextendedtomoduleorderings."ThegF rstpartofthissectionisdevotedtoexplainingtheseconceptsandto gettingabGetterinsightintotheirmathematicalmeaning.ThenweaddressaveryimpGortantproblem.OneofthemainideasinComputationalCommu-tativewAlgebraistostudyordetectpropGertiesofidealsandmodulesusingtheinformationscomingfromtheirassoGciatedleadingtermidealsandmodules.Theqreasonisthatthelatterob8jects,whosenatureispurelycombinatorial,areeasiertodealwith,andthe rststepinthisdirectionisMacaulay'sBasisTheorem."SuppGosewehaveanidealIinapGolynomialringP*=K[x1|s;:::;xnq~]overa. eld.K.ItisclearthattheresidueclassringPV=IPcanbGeviewedasa궵K-vector~space.Somenaturalquestionsarise.IsitpGossibletoexhibitanexplicitbasis?Canwecomputeit?Thesecondquestionwilltakemuchmoree ort,;lbutwiththeaidofleadingterms,Macaulay'sBasisTheoremyieldsabGeautiful3@answertothe rstone.ThistheoremrequirestheassumptionthattheQmoGduleorderingisatermordering.Thuswesee,forthe rsttime,thetheoreticalUUimpGortanceoftermorderings."Inہthe nalpartofthissectionweshowhowtwofundamentaltermor-derings,1Lex#and1DegRevLex4q޲,1canbGecharacterizedusingthekindofleadingtermsUUtheyproGduce.<@Wō>;604`1.F:oundations-ō>;"궲In[whatfollows,welet[õRobGearing,nϸ1,P5^=RDz[x1|s;:::;xnq~]a[pGolyno- mialring,r51,andJamoGduleorderingonthesetoftermsT^nq~he1|s;:::;ermi궲ofUUPc^r1.UUThestandardbasisofPc^r &willbGedenotedbyfe1|s;:::;ermgasusual.RemarkT1.5.1._uHEveryNelementNm2Pc^r>nlf0ghasNauniquerepresentationasaUUlinearcombinationofterms8m=s X tմi=1㉵ciTLtie iLt궲wherec1|s;:::;cs R2RnXf0g,t1;:::;ts R2T^nq~, 1;:::; s R2f1;:::;rGg,andwhere궵t1|se 1 a>)t2e 2>)m>)tsF:e s ."Ifwewritemn=f1|se1+߲+frmerm,wheref1;:::;fr2nPc,thenwehaveSupp*#(fiTL)=ftjĸj j=igUUforUUeachi2f1;:::;rGg.De nitionT1.5.2.is~F*or]anon-zeroelement]m2Pc^r1,]letm=Pލ USs% USi=1tJciTLtie i ΪbGetheUUrepresentationaccordingtoRemark1.5.1.a)%The#term#L*TFH(m)H=t1|se 1 92T^nq~he1;:::;ermi#is#calledtheleading.term%ofUUmwithUUrespGecttoUU[ٲ.?b)%The element LC"(m)TH=c1л2R!nqZf0g iscalledtheleading6coQecien9t%ofLmwithLrespGecttoL[ٲ.IfLCŒ(((m)=1,LwesaythatLmis[-monic,Lor%simplyUUmonicifUU.isclearfromthecontext.\Ic)%W*eUUletUULM"(m)=LC?(m)8LTo?(m)=c1|st1e 1 u."F*orthezerovectorm*=(0;:::;0),werecallfromDe nition1.1.11thatSupp*#(m)=;.مTheleadingtermمL*T}ߪ(m)مandtheleadingcoGecientLCRj(m)areUUnotde ned."Note thattheleadingtermofavector m2Pc^rnf0greally consistsoftwoPdata:thetermPt1 +29T^n >βwhichissometimesalsocalledtheleadingpQo9werproduct:of:m,andthepGosition 123@f1;:::;rgof:thistermwhichissometimescalledtheleadingL,pQositionofm.InC0oLCoA,theleadingpGowerproGductrofavectorcanbeobtainedusingthefunctionrLPP2(::: ),andtheleadingUUpGositionisaccessibleviaUULPosUI(::: )."With:respGecttotheusualoperationssuchasadditionandmultiplicationofpGolynomials,leadingtermsbehaveprettymuchasonewouldexpGect:theleadingetermofasumisthebiggestleadingtermofoneofthesummands,exceptdPifsome\cancellation"oGccurs,andtheleadingtermoftheproductisthepproGductoftheleadingterms,exceptforsomepathologicalcases.LetuscollectUUthepreciserules.PropQositionT1.5.3.q(RulesTforComputingwithLeadingT erms)Asab}'ove,weletP*=RDz[x1|s;:::;xnq~]beapolynomialringoveraringRaandaVHmo}'duleorderingonVHT^nq~he1|s;:::;ermi.Moreover,letVHfV;f1|s;f2C2PbeVHnon-zerop}'olynomials,andletm;m1|s;m2C2Pc^r ebenon-zerovectorsofpolynomials.=ՠWō>;1.5LeadingTT:ermsٮ61-ō>;~oa)%We have SuppB(m1[+Gm2|s)lSupp(m1)G[Supp(m2), andifmor}'eover %m1S+8m2C6=0,thenL*T7v (m1+m2|s))max$ (fL*T %(m1);L*TN7Ͳ(m2)g.Hb)%Supp}'osethatm1Բ+am22N6=0,andsupposethatL*TBز(m1|s)۸6=L*TYj(m2)or%LC3*8(m1|s)8+LCş[(m2)6=0.ThenwehavekHbIL*Tn؟t n(m1S+8m2|s)=maxc$fL*T %(m1);L*TN7Ͳ(m2)gHc)%F;ort2T^nq~,wehaveL*T7v (tm)=t8L*To?(m). ~od)%IfR/cisaninte}'graldomain,andiftistheterminSuppT(f)forwhich%t8L*To?(m)ismaximalwithr}'espectto[,thenL*T7v (fm)=t8L*To?(m).He)%If:Ris:aninte}'gral:domain,andifisamonoidor}'deringonT^n esuch%thatisc}'ompatiblewith!,thenwehavekHL*TZ?(fm)=L*Tjiܲ(f)8L*To?(m)%InWp}'articular,ifWݵRkisanintegraldomain,thenWݲL*Tl(f1|sf2)*=L*Tͥڲ(f1|s)߸ %L*T2U.7Tc(f2|s).Pr}'oof.6T*oprovea),writem1̲=YPލs%i=1ŋciTLtie i andm2̲=YPލsr0%jg=1c^0;Zj6t^0;Zje n908j fac-cordingUUtoRemark1.5.1.F*romtherepresentationࠍ@ km1S+8m2C= ,r X tմi=1!X +㉴t2Tn&*`?#X j.,fjgjtja=t; j=igaRcjo+MX jfjgjt08ja=t; n908j=ig5^+c0፴j6` /tei!Ӎ궲we_concludethat_Supp(m1g+꡵m2|s){Suppc(m1)꡸[Supp#(m2)_and_alsothat궵teid)max$ (ft1|se 1u;t^0l1e n901gUUforUUallteid2Supp(m1S+8m2|s)."F*orDtheproGofofb),werepresentDm1|s,m2 andm1+m2asDabGove.IfweOhaveOL*TV(m1|s)h=t1e 1 A=t^0l1e n901 A=L*T [n(m2),OthenOc1\h+c^0l1?6=0impliesthatL*TD7(m1Ѳ+k^m2|s)ER=t1e 1 Dz=maxȟD^ft1e 1u;t^0l1e n901g.Whent1e 1 ǵ<t^0l1e n901 ;or궵t1|se 1 a>)t^0l1e n901 u,COtheclaimfollowsimmediatelyfromtheabGoverepresentationofUUm1S+8m2|s."Inordertoshowclaimc),wewritem=Pލ USs% USi=1tJciTLtie i GasinRemark1.5.1.Then~tm d=Pލ s% i=1ciTL(tti)e i ]6is~therepresentationoftm,sincetiTLe i 걵>ntj6e j궲for1i㗲(ttj6)e j o.ThusweobtainL*Thp(tm)=궵tt1|se 1 a=t8L*To?(m).="F*orqtheproGofofd),werepresentqfղ=FPލ6s%6i=1UxciTLti0andmF=Pލ6sr0%6jg=17صc^0;Zj6t^0;Zje j궲according-toRemark1.5.1.Thenwehave-tiTLt^0;Zj6e j 6θ)tiL*T(m)tL*TN7Ͳ(m)for{iV=1;:::;sand{forj=1;:::;s^09.{Letck  bGethecoecientof{tinf. ĔNowUUtheclaimfollowsfromUUfm=Pލ USs% USi=1tJPލ'sr0%'jg=15Y4(ciTLc^0;Zj6)(tit^0;Zj)e j andUUck됵c^0l1C6=0."T*oprovee),weobservethatifziscompatiblewith!Dz,thentheterm궵t=L*Tjiܲ(f)istheuniqueelementofSupp˛(f)forwhichtL*T W-ò(m)ismaximalwithUUrespGecttoUU[ٲ.kHDe nitionT1.5.4.is~LetUUM3Pc^r &bGeUUaPc-submodule.a)%ThemoGduleL*TM-ò(M)S=hL*T %(m)jm2M+nqf0giiscalledtheleading%termTmoQduleUUofUUMlpwithrespGectto[ٲ.>Wō>;624`1.F:oundations-ō>;?ֲb)%If:r=z1,:i.e.ifMzPc,thentheidealL*Tdɟ_(M)zP$ɲis:alsocalledthe %leadingTtermidealUUofUUMlpwithrespGectto[ٲ.\Ic)%ThemonomoGdulefL*T %(m)jm2M'n f0ggT^nq~he1|s;:::;ermiwillbGe%denotedUUbyUUL*T[zfMg.q"Noticethat,forM(@=%h0i,wegetL*TX(M)%=h0iandL*TXfMg=;궲using*thisde nition.If*m1|s;:::;ms m2&̵Pc^r [arenon-zerovectors,andif궵Mr)=[hm1|s;:::;msF:iPc^r isQthesubmoGdulegeneratedbythem,wehave궸hL*T %(m1|s);:::;L*TN7Ͳ(msF:)iL*Tj=(M).=8ThefollowingexampleshowsthatthiscanUUbGeaproperinclusion.ExampleT1.5.5.cZ1.3.9.aimpliesthat>ZL*TDfMg>Zisgeneratedby nitelymanytermsLasamonomoGduleoverLT^nq~.LThusthosetermsgeneratetheLRDz-moGduleL*T E%۲(M),UUandusinga)wegetclaimb).MҪ"궲NowD5wearereadytoprovethemainresultofthissection.AswesaidbGefore,wMacaulay'sBasisTheoremrequirestheassumptionthatthemoduleorderinglisatermordering.F*urthermore,weneedtoassumethatourbaseringUUisa eld.TheoremT1.5.7.dH(Macaula9y'sTBasisTheorem)L}'etKbea eld,letPi=aڵK[x1|s;:::;xnq~]b}'eapolynomialringoverK,letMԸӹPc^r b}'eaP-submodule,andletZbeamoduletermorderingonT^nq~he1|s;:::;ermi.7Wedenotethesetofalltermsin7T^nhe1|s;:::;erminL*TS>ԸfMgbyڵBq.Thenther}'esidueclassesoftheelementsofB7Kformab}'asisofthe궵K-ve}'ctorspacePc^r1=qM.Pr}'oof.6Firstzweprovethattheelements\qzzb θ2Pc^r1=qM앲suchthatzb2BUformassystemofgeneratorsofsPc^r1=qM.Inotherwords,weneedtoprovethatthevectorsubspaceN3=P USb2BڵKnb+MöequalsPc^r1.F*oracontradictionsuppGose?Wō>;1.5LeadingTT:ermsٮ63-ō>;궲that8|N3Pc^r1.8|ThenthesetPc^r`n/NOcontainssomenon-zeroelements.Hence TheoremE1.4.19impliesthatthereexistsanelementE۵mofPc^rjn9N\havinga4minimalleadingtermwithrespGectto4[ٲ.IfnowL*T<:Ҳ(m);T2Bq,4thentheelementmLC/5(m)L*TN7Ͳ(m)isstillinPc^rnN)8andhasasmallerleadingtermthan銵m:acontradiction.ThusweneedtohaveL*Tﯲ(m)2L*Ta@fMg.Sothereexistsanelementm^0Т2iM)suchthatL*T(m^09)i=L*T(m).Again theelementеm&hLCj(m)ʉfeuLC hv(m0s)$hsm^0 liesinPc^r]nNandhasasmallerleadingterm $thanUUm:UUacontradictionagain."NowweprovelinearindepGendence.Supposethereisarelationm=궟Pލxs%xi=1-ciTLmid2M suchthatc1|s;:::;cs R2KS8nf0gandm1|s;:::;ms R2Bq.ThenwehavehL*T%(m)2L*Tj=fMg,hsincehm2M.W*ealsohavehL*T%(m)2Supp(m)fm1|s;:::;msF:g>eBq,bGecausem1;:::;ms$areterms,andbGecauseofProposi-tionn1.5.3.a.Altogetherwe ndnL*T(m)2L*Tj=fMg\BG=;nwhichnisimpGos-sible.-oҬ"궲T*oseehowessentialtheassumptionisthat$isatermordering,considertheUUfollowingexample. ɍExampleT1.5.8.cfMg.)Unfortunately*,wedonotyetknowhowtocalculate L*Tk(M), andwecannotstorethebasis ܸf\qUb¸jb2Bqgin acomputer,since{itisingeneralin nite.InthenextchapterweshallseehowtoovercometheseUUproblems."T*o'Gconcludethissection,weshowhowtocharacterizetwoofthemostimpGortant!0termorderingson!0T^n=Z,namely!0LexWand!0DegRevLex9EintermsoftheirfqbGehaviourwhentheyareusedtoorderpolynomials.So,fortherestoftheUUsection,letUURibGearing,andletP*=RDz[x1|s;:::;xnq~].PropQositionT1.5.10.wL}'etbeamonoidor}'deringonT^nq~.Thenthefollowingc}'onditionsareequivalent.~oa)&"=LexHb)%F;orfڧ2P_andi2f1;:::;ngsuchthatL*TC(f)2RDz[xiTL;:::;xnq~],wehave%fڧ2RDz[xiTL;:::;xnq~].Pr}'oof.6T*obproveba)Ը)b),letbfc2PƈbGesuchthatbL*TLex(f)2RDz[xiTL;:::;xnq~].Ifiat2Supp(f)`nfL*T LexK(f)g,iathenbyDe nition1.4.3the rstnon-zerocompGo-nentoflogF(L*T LexK(f))mlog (t)ispGositive.Butsincethe rstim1compGonentsof*L*TΉLexv(f)*are*zero,alsothe rstiN1compGonents*oflog(t)have*tobGezero.UUThuswegetUUt2RDz[xiTL;:::;xnq~]forUUallt2Supp(f).@쩠Wō>;644`1.F:oundations-ō>;"궲Nowweproveb)k)a).Letusconsidertwotermst1|s;t2 {2kT^n \esuch thattlog (t1|s)HIlog2(t2)m=(0;:::;0;ci1 ;:::;cnq~),twithtci1^>m0.The rst궵iX2coGordinatesoflog5(t1|s)andlog(t2|s)areequal,sowecanusepropGertyd)6ofDe nition1.4.1andassumethattheyarezero.F*orthesamereasonwehcanalsoassumethattheh(i1)^st 3$coGordinatehoflogR(t2|s)iszero,whilethe(i1)^st coGordinateoflog(t1|s)isdi erentfromzero.Nextweconsiderthe5pGolynomial5fPQ=<µt1Jʲ+Wt2|s.Supposeforcontradictionthat5L*T;(f)<=t2|s.Then?L*TUE(f)MҸ2RDz[xiTL;:::;xnq~],?andb)impliesthatalso?Ƶfaa2MҵR[xiTL;:::;xnq~].Consequently*,aweseethatat1W=ۇfTA t22RDz[xiTL;:::;xnq~],aincontradictionwiththe@factthatthe@(iJ1)^st coGordinate@oflog+4(t1|s)isdi erentfromzero.ThereforeweUUhaveUUL*T[z(f)=t1|s,i.e.UUt1C>)t2."Altogether,& wehaveshownthat& t1`>Lext2 |impliest1> t2|s.& Byinter-changingqt1_andt2|s,qwe ndthatqt1C>Lexo!t2_ifandonlyift1C>)t2|s.ThereforeweUUhaveUU"=Lex.hePropQositionT1.5.11.wL}'etbeamonoidor}'deringonT^nq~.Thenthefollowingc}'onditionsareequivalent.~oa)&"=RevLexHb)%F;orKffڸ2SKPDandi2f1;:::;ngsuchKthatL*Tڟp(f)2(xiTL;:::;xnq~),Kwehave%fڧ2(xiTL;:::;xnq~).Pr}'oof.6T*o#prove#a) )b),let#f22PCbGesuchthat#L*TC)ٲ(f)#isintheideal(xiTL;:::;xnq~).Ift2Supp(f).nfL*T %(f)g,thenbythede nitionofRevLex,thelastDnon-zerocompGonentofDlog(L*T %(f)))log(t)DisDnegative.Butsincethelastnon-zerocompGonentoflogr(L*T %(f))isinapositionbetweenƵiandn,also$thelastnon-zerocompGonentof$log(t)$hastobeinapositionbetween궵iandn.ThismeansthatallthetermsinSupp(f)havetobGeintheideal(xiTL;:::;xnq~)."Now~weprove~b))a).Let~t1|s;t2|bGetwotermsin~T^n qsuchthat~log*(t1|s)3궲log b(t2|s)=(c1;:::;ciTL;0;:::;0),ewitheci<0.ThelastenRicoGordinateseoflog b(t1|s)oandlog(t2)areoequal,sowecanusepropGertyd)ofDe nition1.4.1andassumethattheyarezero.F*orthesamereasonwecanalsoassumethatthe5/i^th ?ܲcoGordinate5/oflog(t1|s)iszerowhilethei^th ?ܲcoGordinateoflog(t2|s)isdi erentYYfromzero.LetusconsiderthepGolynomialYYfX=ɵt1+;t2|s.SupposeforcontradictionthatL*T-P(f)v=t2|s.ThenL*T-P(f)2(xiTL;:::;xnq~),andb)impliesthatalsoĵf 2z(xiTL;:::;xnq~).Consequently*,wehaveĵt1=zfcԵt22(xiTL;:::;xnq~),in9contradictionwiththefactthatthelast9nixicoGordinates9oflog(t1|s)arezero. Therefore L*TAײ(f)\=t1|s,i.e.wehave t1qϵ> Wt2|s,andwemayconcludethatUU"=RevLex#G."궲LaterooninthisChapter(seeSection1.7)andinthesecondvolumewewillstudytheconceptsofgradingsandhomogeneityingreatdetail.However,formthemomentitisenoughtorecallthatapGolynomialofdegreemdismsaidtoebGehomogeneousifallthetermsinitssupporthaveedegreeed.Also,werefertoDe nition1.4.9forthenotionofdegree-compatibletermorderings.AWō>;1.5LeadingTT:ermsٮ65-ō>;궲Then aneasymoGdi cationoftheproofoftheprecedingpropositionyieldsthe followingUUcharacterizationofthedegree-reverse-lexicographictermordering.uCorollaryT1.5.12.lL}'etWsbeade}'gree-compatibletermor}'deringonT^nq~.Thenthefollowingc}'onditionsareequivalent.~oa)&"=DegRevLexHb)%F;ordgeveryhomo}'geneousdgpolynomialdgfڧ2Pandeveryi2f1;:::;ngdgsuch%thatL*T7v (f)2(xiTL;:::;xnq~),wehavefڧ2(xi;:::;xnq~).)%Exercise1.tLet+Uf=sx-=2h1G+x1*x2+x-=2h22sPV=K[x1*;x2].+USho9wthatthere %isTnomonoidorderingTgonT-=2UOsuc9hTthatL:Tɿ(f)=x1*x2*.F%Exerciseɏ2.VF:or eac9hofthefollowingpAolynomialsin Q[x1*;x2;x3], nd%a{termorderingsuc9hthatthegivenrepresentationagreeswiththeone%pro9videdTbyRemark1.5.1. *?a)8xf1m=x1*x-=3h2x38+82x1x-=2h2x-=2h38x-=2h1x-=3h3)b)8xf2m=4x-=4h1*x-=5h2x38+82x-=3h1x-=2h2x3+8x1x-=2h2x-=4h3*6c)8xf3m=x-=2h1*x-=4h2x38+83x1x-=6h282x-=2h1x-=2h2%Exercise@3.DoktheleadingtermsofthepAolynomialskf1Ly=!x-=4h1*x2r}Gx-=5h3*,%f2m=x-=3h1*x-=3h2/$}1,wandwf3=x-=2h1*x-=4h2/$}2x3;generatewtheleadingtermidealwith%respAectTtoTDegRevLex2oftheidealT(f1*;f2;f3)TinQ[x1*;x2;x3]?%Exercise 4.G Let7KbAe7a eld.T:rytouseMacaula9y'sBasisTheoremto%determine^anexplicit^K-basisoftheringK[x1*;x2;x3]=(x1Lx-=3h3;x2Lx-=4h3).%Hint:TUsethelexicographictermordering.%ExerciseK5.0LettnKVbAetna eld,letPy)=0FK[x1*;::: ;xn7],let10Fm;664`1.F:oundations-ō>;T utorialT11:P9olynomialRepresentationIQI򌍑궲UsingRemark1.5.1,wehaveanotherpGossiblewayofrepresentingpGolynomi- alsfromthepGolynomialringP*=K[x1|s;:::;xnq~]overa eldKsinacomputerprogram.FW*echoGoseatermorderingF]onT^nq~.FForeachFfڧ2P~͸n>f0g,welet[ǵfڧ=s X tմi=1㉵ciTLti궲with;c1|s;:::;cs R2Knif0gand;witht1;:::;ts R2T^n such;thatt1C>)m>)ts궲bGe=therepresentationaccordingtoRemark1.5.1.Thenwerepresent=f̲inthecomputerUUprogrambythelistofpairs/}@[[c1|s;logT(t1)];:::;[csF:;log(tsF:)]]whereUUlog@(t1|s);:::;logT(tsF:)UUareUUconsideredasvectorsinZ^nq~.a)%W*riteߵaC0oLCoA$aprogramߵReprPoly25(::: )ߵwhichtakesapGolynomialin %Q[x1|s;:::;xnq~]UUandUUcomputesthisrepresentation.?b)%ImplementC0oLCoA$zfunctionsAddPoly2/(::: )andMultPoly24꽲(:::)which%calculatevthelistscorrespGondingtothesumsandproductsoftwovpoly-%nomialsUUrepresentedinthisway*.\Ic)%Check(thecorrectnessofyourprogramsbyapplyingthemtothepGoly-%nomialsofT*utorial1.f.Computethelistsrepresentingf1퇲+qf2|s,f1qf2|s,%andUUf2S8f3+f^3l1 WagainUUintwoUUways.&T utorialT12:SymmetricP9olynomials궲LettKbGeta eldandf2P聲=K[x1|s;:::;xnq~]atpGolynomial.W*ecallfsymmetric +if +f3isinvqariant +underallpGermutationsoftheindeterminates궵x1|s;:::;xnq~.UUF*orUUi=1;:::;n,UUthepGolynomials sid= 6`X tj1 < ;1.5LeadingTT:ermsٮ67-ō>;\Ic)%Prove}therecursiveformula}sid=-R~si (+]2xn׸~q~si1forn>1}andi2Z,}where &~%s1.;:::;~sn19\are0theelementarysymmetricpGolynomialsintheindeter-%minates/x1|s;:::;xn1,/wherewesetsiy=g~[-si=[-0ifi<0ori>n,/and%whereUUs0C=-R~s0=1.?d)%Usemc)towriteaC0oLCoA$MprogrammElSym^(::: )mwhichcomputesthei^th%elementaryUUsymmetricpGolynomialinUUnindeterminates.\Ie)%ProvethattheleadingtermL*T}Lex%(f)=x: 1l1 Dx^ n፴nt?ofasymmetricpGoly-%nomialUUfڧ2Pon8f0gsatis es 1C8 nq~.f)%Show٩thatonecansubtractfrom٩f8asuitablemultipleofthepGolynomial s%s: 1 2l1's㍴ n1 ; nÍn1%s^ n፴n΋such4=thattheresultisasymmetricpGolynomial%withvDasmallerleadingtermwithrespGecttovDLex6;.Consequently*,develop%anUUalgorithmforrepresentingUUfhasapGolynomialins1|s;:::;snq~.g)%ImplementsIthealgorithmfromf)inaC0oLCoA functionsIReprSym(34(::: )sIwhich%takesapGolynomialʵf"2P}Yandreturnsapolynomialʵgj\2P}Ysuchthat%fڧ=g[ٲ(s1|s;:::;snq~).?h)%Apply|IsffSymmetricB(::: )|andReprSym)og(:::)to|thefollowingpGolynomi-%als.)1)8xF1C=x^3l1|sx2S+8x^3l2x3+8x1x^3l2+8x1x^3l3+8x^3l1x3+8x2x^3l3C2Q[x1;x2;x3])2)8xF2C=P USi6=j.x^2;Zi|sxjĸ2Q[x1;:::;x5] M)3)8xF3C=x^5l1S+8g+8x^5l52Q[x1|s;:::;x5] T utorialT13:NewtonP9olytopQes궲Givenanon-zeropGolynomialյfdinapolynomialringյP*=K[x1|s;:::;xnq~]overa eldK,wemaywonderwhichtermsinitssuppGortcanbetheleadingtermwithrespGecttosometermordering.ApartialanswertothisquestioncanbegivenҡusingtheNewton?pQolytopeҡofҡf0whichisthesub8jectofthistutorial."F*or3v1|s;v228R^nq~,3thesetf1|sv1I(+̵2v2j81;228R0 t;1I(+̵2=81g3iscalledtheline segmen9tde nedbyv1 /andv2andisdenotedby[v1|sv2].AsubsetzSR^n iszcalledcon9vexzifforallv1|s;v22Sthelinesegment[v1|sv2]isUUcontainedinUUS.a)%LetaS۸NR^n (bGeaanon-emptysubset.Showthatthereexistsaunique%convexYsubsetofYR^n MײcontainingSowhichYiscontainedineveryother%convexKsetcontainingKS.Itiscalledthecon9vex.hullKofS0anddenoted%byUUconvx(S).%Hint:UUConsidertheintersectionofallconvexsetscontainingUUS.?b)%LetBS漸S/R^n |bGeBaconvexBset,andletv2S/R^nHθnPS.ProveBtheequality%conv9/(Sv[%fv[ٸg)=f[vwD]jw\2Sg,butalsogiveanexampleofaset%S^0(޸R^n ӲsuchUUthatUUf[v[wD]jv;w 2S^0aƸgUUisUUnottheconvexUUhullofUUS^0aƲ."F*orda nitesubsetdS=)fv1|s;:::;vrmgofR^nq~,ditsconvexdhulldP=convM(S)is2alsocalledapQolytope.2Av9ertexof2Pﯲisanelementvp 22Pﯲsuch2that궵v?c="2 conv E(P ]n8fv[ٸg).UUThesetofallverticesofUUP'ҲisdenotedbyUUV*ert@(P}).D<=Wō>;684`1.F:oundations-ō>;\Ic)%ShowUUthatUUP=fPލ ;r% ;i=12iTLvidj1|s;:::;r42R0 t;1S+8g+8r4=1g. %Hint:AUseinductiononArG,thefactthatifwelet n=PލX/r71%X/i=1iTL,then%Pލ0?ڴr%0?i=1?^ѵiTLvid= z(Pލ ;r71% ;i=1h,i,ʉfe2[( %2vi)8+rmvrm,UUandapplyb).?d)%ShowUUthatUUV*ert@(P})S. \Ie)%Prove{>thattheconvex{>hullof{>V*erte(P}){>isPMandthat,amongallsets%whose@convexhullis@P},itistheuniqueminimalsetwiththatpropGerty*."Nowletusreturntoournon-zeropGolynomialȵfڧ2P*=K[x1|s;:::;xnq~].W*eletS[=θflog (t)jt2SuppS(f)gandcallNewton&$(f)=conv>^(S)theNewtonpQolytopeoff.F*urther,weletV*ertt-(f)bGethesubsetofSuppi(f)whichcorrespGondsUUtothesetofverticesofUUNewton%(f).f)%ProveNthatifNt2Supp(f)*VnV*ert(f),NthenthereisnomonoidorderingN%suchUUthatUUL*T[z(f)=t.%Hint:βUsethefactthatifεS=7fv1|s;:::;vrmgQ^nq~,theneveryelementεv%ofP#\Q^n z}hasarepresentationvN_=Pލr%i=1iTLvi]Kwith1|s;:::;r`(2Q0%andV1KW+ Ho+r4=1.V(Y*oudonothaveVtoproveVthis.)NowletV*ert(f)=%ft1|s;:::;tsF:g.(Findarelation(log(t)=Pލ USs% USi=1tJilog(tiTL)(withid2Q0 t,(and%thusarelation̵t^a0 =Qލ 8s% 8i=1Wصt:ai;Zi 4witha0|s;:::;as R2Nanda0C=a1>C+и .G+еasF:.g)%LetUUfڧ=3x^6|sy[ٟ^2,+82x^3y[ٟ^38xy+5x^3|sy[ٟ^5d2K[x;y[ٲ].)1)7xFindUUatermorderingUU.suchthatL*T[z(f)=x^6|sy[ٟ^2L.)2)7xFindUUatermorderingUU.suchthatL*T[z(f)=x^3|sy[ٟ^5L.)3)7xShowUUthatUUL*T[z(f)6=xy.foreverytermorderingUU[ٲ.)4)7xShowUUthatUUL*T[z(f)6=x^3|sy[ٟ^3 -foreverymonoidorderingUU[ٲ.EQӠWō>;V1.6TheTDivisionAlgorithmٮ69-ō>;1.6**TheDivisionAlgorithm(&DivideN;704`1.F:oundations-ō>; Tʱ3Tʟ&fes2Ypx2 |k+ 3+33+&fes4 bѵxSs _feE̻ w@1w@&fes4|3x+1 w@1w@&fes4|3x+ 3+13+&fes8u  _fe$C܎ C7C&fes8ɍ궲In;V1.6TheTDivisionAlgorithmٮ71-ō>;궲Inotherwords,we ndarepresentationfڧ=q^[ٷ0l1|sg^[ٷ0l1A+{q^[ٷ0l2g^[ٷ0l2+{p^0Q=q^[ٷ0l2g1+{q^[ٷ0l1g2+{p^0 궲suchUUthatUUq^[ٷ0l1C=x1S+81,q^[ٷ0l2=x1|s,UUandp^0Q=2x1S+81."TheproGceduredescribedabovecanbeextendedtoaverygeneralsit-uation.6Itprovidesuswiththefollowingalgorithm.Notethatwhenever궵t;t^09;t^00 z2T^n esatisfy`t=t^0t^00r,`andforall`i2f1;:::;rGg,`weshallcommitaUUslightabuseofnotationandwriteUUt^00㊲=hTteiKʉfe t0sei݊.TheoremT1.6.4.dH(TheTDivisionAlgorithm)L}'ets֭1,andletm;g1|s;:::;gs2֭Pc^rMn?f0g.Considerthefollowingse}'quenceofinstructions.~o1)%L}'etq1C=8ײ=qs R=0,p=0,andv"=m.~o2)%Find)thesmallest)icu2f1;:::;sgsuch)thatL*T5N(v[ٲ)isamultipleof %L*T2U.7IJ(giTL).zIfsuchanziexists,zr}'eplacezqi byqi"+&hcLMNH(v@L) ʉfe6LM #(gi*)+TandvSbyLc%v&hƱLMX(v@L)lʉfe6LM #(gi*)%ĸ8giTL. ˍ~o3)%R}'epeat-step2)untilther}'eisnomore-i2f1;:::;sgsuch-thatL*TR(v[ٲ)isa%multipleofL*T^J(giTL).Thenr}'eplacepbyp`Ų+LMr.(v[ٲ)andvbyv`ŲLMr(v[ٲ).~o4)%If;=now;=v!׸6=0,startagainwithstep2).Ifv!ײ=0,r}'eturnthetuple%(q1|s;:::;qsF:)2Pc^s =andtheve}'ctorp2Pc^r1.Thisisanalgorithmwhichr}'eturnsvectors˲(q1|s;:::;qsF:)x2Pc^s >andp2Pc^rsuchthatm=q1|sg1S+8g+8qsF:gs+pandsuchthatthefollowingc}'onditionsaresatis ed.~oa)%NoelementofSupp(p)isc}'ontainedinhL*T %(g1|s);:::;L*TN7Ͳ(gsF:)i.Hb)%Ifqid6=0forsomei2f1;:::;sg,thenwehaveL*T7v (qiTLgi))L*T=/Ӳ(m).Hc)%F;or]allindic}'es]ʵi=1;:::;sand]alltermstinthesupp}'ortofqiTL,wehave%t8L*To?(giTL)㊵=2 8hL*T %(g1|s);:::;L*TN7Ͳ(gi1 )i."Mor}'eover,theve}'ctorsֲ(q1|s;:::;qsF:)@Y2Pc^s andp2Pc^r satisfyingtheab}'ovec}'onditionsareuniquelydeterminedbythetuple(m;g1|s;:::;gsF:)2(Pc^r1)^s+1f=.Pr}'oof.6FirstPzweobservethatateachpGointintheDivisionAlgorithmtheequationym=q1|sg1S+8g+8qsF:gs+p+v}궲holds,Esinceinstep2)wehaveEqiTLgil+iv"=(qi+&hOLM8(v@L)Kʉfe6LM #(gi*)"m)gi+(vtB&hOLM8(v@L)Kʉfe6LM #(gi*)"mgiTL), ˍandUUinstep3)wehaveUUp8+v"=(p+LM#(v[ٲ))+(vLM(v[ٲ))."The^algorithmstopsafter nitelymanysteps,bGecausebothinstep2)andinstep3)theleadingtermL*T}7(v[ٲ)bGecomesstrictlysmallerwithrespecttoUU[ٲ.UUByTheorem1.4.19,thiscanhappGenonly nitelymanytimes."WhenQthealgorithmstops,wehaveQm=q1|sg1#M+ڸ [+ڵqsF:gs+p.QThevectorQp궲satis esJpropGertya),sinceinstep3)ascalarmultipleofatermisaddedtoJp궲only)qifthattermisnotamultipleofoneoftheterms)qL*T/(g1|s);:::;L*TN7Ͳ(gsF:)."NowmweprovebyinductiononthenumbGerofstepsprocessedthatwealways have L*T}(v[ٲ)k ZL*T`&(m) andL*T}(qiTLgi)k ZL*T`&(m) whenqiK6=0.HWō>;724`1.F:oundations-ō>;궲Thisisobviouslysatis edatthestartofthealgorithm.Everytimestep2) isexecutedandtheoldandnewvqaluesofqi ?Narenotzero,wehavetheinequalities _&S[L*T2:(b>(qi,+&hƱLMX(v@L)lʉfe6LM #(gi*)")8giTLb wӸ Ʋmax<"ҸfL*T %(qiTLgi);L*TN7Ͳ(v[ٲ)g)L*T=/Ӳ(m):$궲Thesameconclusionholdstriviallyiftheoldvqalueofqi5waszero.ThusconditionUUb)continuesUUtoholdthroughoutthealgorithm."F*urthermore,|conditionc)isalways|satis ed,sinceinstep2)onlyscalarmultiplesoftermst2T^nq~he1|s;:::;ermiareaddedtoqi3jforwhichtL*T8J(giTL)wasnoteliminatedfromɵv뢲duringanearlierexecutionofstep2),i.e.whichareUUnotamultipleofoneofthetermsUUL*T[z(g1|s);:::;L*TN7Ͳ(gi1 )."Finally*,>weshowuniqueness.SuppGosetherearetworepresentations>m=궵q1|sg1+$9+$9qsF:gsjs+p=q^[ٷ0l1g1+$9+$9q^[ٷ0፴sF:gsjs+p^0;whichKsatisfyconditionsa),b),andUUc).ThenwehaveR^0=(q1S8q^[ٷ0l1|s)g1+g+(qsq^[ٷ0፴sF:)gs+(pp^09)?t9()"Condition9a)impliesthat9L*Tȟ^(pV"p^09)=<2$hL*T %(g1|s);:::;L*TN7Ͳ(gsF:)i,9and conditionfc)impliesthatfL*T^((qi{'q^[ٷ0;ZiTL)gi)8O=ݸ2fhL*T %(g1|s);:::;L*TN7Ͳ(gi1 )ifforallT9io2f1;:::;sgwithqi36=q^[ٷ0;ZiTL.T9Thustheleadingtermsofthesummandsin~()are~pairwisedi erent.InviewofRule1.5.3.b,thisisimpGossibleunless궵q1S8q^[ٷ0l1C=8ײ=qsq^[ٷ0፴s R=pp^0Q=0.n!RemarkT1.6.5._uHUsingtheDivisionAlgorithm,itisnotalwayspGossibleto [decidewhethertheelement [mnǸ2Pc^r is [containedinthesubmoGd-ule6)hg1|s;:::;gsF:iuPc^r1.6)F*orinstance,ifnu=2,r/=1,PL=Q[x1|s;x2],궵m=x1|sx^2l2:"x1|s,g1=x1x2:"+1,andg2=x^2l21,thenwecalculatewithrespGectUUtoUULexjthefollowingrepresentation:q΍;꺵x1|sx22 |kx1 =0^ٵg1S8(x2)ٵg2S8(0)H remainder6k8x1Sx2#ፍ;꺵x1|sx22 |k+x2;꺟tfe0aHW)x1 |kx2궲ThusMwe ndMڵm=q1|sg1a+)q2g2+)pMڲwithq1C=x2|s,q2=0,Mandp=x1a)x2C6=0.But,UUinfact,theelementUUm=x1S8g2ȲisUUintheideal(g1|s;g2)Pc."TheyDivisionAlgorithmallowsustoexpresstheresidueclassofanelement#صmmoGdulo#thesubmodulegeneratedby#ظfg1|s;:::;gsF:gas#alin-earm3combinationofthosetermswhicharenotmultiplesofanytermin궸fL*T %(g1|s);:::;L*TN7Ͳ(gsF:)g.ButthesetofthosetermsisingeneralnotthedesiredUUbasisofUUPc^r1=qM,asthefollowingexampleshows.I۠Wō>;V1.6TheTDivisionAlgorithmٮ73-ō>;ExampleT1.6.6.cf0g,RandletGղbGethetuple(g1|s;:::;gsF:).W*eapplytheDivisionAlgorithmandobtainarepresen-tationhFm檲=q1|sg1+E5+EqsF:gs+pwithq1|s;:::;qs,2檵Pղandp2Pc^r1.hFThenthevectorٵpiscalledthenormallremainderofٵmwithrespGecttoٸGandisdenotedbyNR_D;G괲(m),orsimplybyNR_DGW(m)ifnoconfusioncanarise.F*or궵m=0,UUweletUUNR1ȟG۲(m)=0."In otherpublications,thenormalremainderofavectorissometimesalsocalledVLitsnormalformwithrespGecttoVLGѲ.However,VLweshallreservethelatterUUnotionforamorespGecialsituation(seeSection2.4).荑%Exercise|]1.սLetn=2,letPک=K[x;yR],andletp=DegRevLex. .Apply %theTDivisionAlgorithmtodivideTfb9y(g1*;g2)TinTthefollowingcases. *?a)8xfq=x-=28+8yR-=2}Q,Tg1m=xy`1,Tg2m=x-=28xy)b)8xfq=x-=7881,Tg1m=x-=28yR,Tg2m=yR-=2x*6c)8xfq=x-=3*yR-=38x-=38yR-=3}Q,Tg1m=xyR-=2x-=2*,Tg2m=x-=2*y`yR-=2 ?%ExerciseA?2.rLet8Pک=K[x1*;::: ;xn7],8letbAeatermorderingonT-=n7,let%g626PH-=rn/2f0g,FandletFM=hgRibAeFthecyclicsubmoduleofFPH-=r generated%b9yTgR.*?a)7xPro9veTthatTL:Tɿ(M)=hL:T k(gR)i.)b)7xSho9wtthattheresidueclassesofthetermscontainedintheset7xT-=n7he1*;::: ;er,pi8nftL:T £(gR)jt2T-=ngTformTaK-basisofPH-=ruS=|rM.*6c)7xConcludeTthatTdimK(PH-=ruS=|rM)=1TifrӍ>1.)d)7xSho9w{that,forevery{m2PH-=ruS,{theDivisionAlgorithmyieldstheunique7xrepresen9tationoftheresidueclassofmmoAduloMgintermsofthe7xbasisTinb). ?%Exercise3.\Letfw;g1*;g2l2BP=K[x1;x2]bAepolynomialssuc9hthat%g1m2K[x1*]andg22K[x2*].Thensho9wthatNRpifh(g'Ӱ1 ;g'Ӱ2))(f)=NRPRfh(g'Ӱ2 ;g'Ӱ1)'(f).%ExerciseR4.xGiv9enanexampleoffourpAolynomialsnfw;g1*;g2;g3m2Q[x;yR;zc]%and atermordering `Msuc9hthatthenormalremainderoffNwithrespAect%toS-G*}=(g1*;g2;g3)S-nev9erS-hasadegree<degG(f),nomatterho9wG$is%ordered.%Exerciser5.Let\fq=yR-=2}Qzc-=2:{Gx-=3x,g1m=yR-=3)x-=21,\andg2m=xyzc-=2%bAepolynomialsinQ[x;yR;zc],andletG/=8(g1*;g2).Giv9eanexampleofa%termorderingNonT-=3Ksuc9hthatNRg0ӎ;G(f)=0andanexampleofaterm%orderingTSonT-=3?suc9hTthatNRN8;G6(f)6=0.JWō>;744`1.F:oundations-ō>;T utorialT14:Implemen9tationoftheDivisionAlgorithm궲In 'thistutorialweconsiderseveralpGossibilitiestoimplementversionsof theWDivisionAlgorithm.AsabGove,WletWK^sbea eld,letWnn1,WletP]=궵K[x1|s;:::;xnq~],~let~r51,let~ZWbGeamoduletermorderingon~T^nq~he1|s;:::;ermi,letUUs1,UUandletg1|s;:::;gs R2Pc^r n8f0g.a)%Program?OaC0oLCoA!function?ODivision.?7(::: )?Owhichtakesanon-zerovector%m|=2Pc^r ,Ѳand[alistofnon-zerovectors[G|==[g1|s;:::;gsF:],[pGerformsthe%DivisionAlgorithm,andcomputesalist[[q1|s;:::;qsF:];p]correspGondingto%the|representation|mqX=q1|sg1l+|+|qsF:gs3+pand|havingpropGertiesa),%b),UUandc)ofTheorem1.6.4.%Hint:F*orimplementingstep2),youmaywanttousetheC0oLCoA BAfunctions%LPP(:::)UUandLPos(:::).?b)%Inthefollowingcases,useDivision/(::: )tocomputerepresentations%asabGove.Inallcases,usebGothPosLexandDegRevLexPos.Checkyour%answersUUbyapplyingthebuilt-inC0oLCoA!ϡfunctionUUDivAlg#C(::: ).)1)8xnm =3,Rr'=2,Rm=(x^2l1^+᧵x^2l2+x^2l3|s;x1x2x3),Rg1}=m (x1;x2),Rg2}=7x(x2|s;x3),UUg3C=(x3;x1))2)8xn=rN=4,|m=(x^4l1|s;x^4l2;x^4l3;x^4l4),|g1=(x1+R1;0;0;0),|g2=(0;x2+7x1;0;0),UUg3C=(0;0;x3S+81;0),UUg4C=(0;0;0;x4S+81))3)8xn=2,r5=5,m=(x^4l1|s;x^3l1x2;x^2l1x^2l2;x1x^3l2;x^4l2),g1C=(x^4l1;x^3l1;x^2l1;x1;1),7xg2C=(1;x2|s;x^2l2;x^3l2;x^4l2)\Ic)%GivengBm;g1|s;:::;gs۸2Pc^rn}f0g,gBconsiderthefollowingsequenceofin-%structions.)1)7xLetUUi=1andq1C=8ײ=qs R=0.)2)7xFindDthelargesttermDܵt2Supp(m)whichDisoftheformt=t^0xL*Tp(giTL)7xfor}some}t^0W2 T^nq~.Ifthereexistssuchaterm,letc 2K ˸nSf0gbGe}its7xcoGecientinm,replacembymj 1pc˟&fe+LC hv(gi*)#5ѵt^0xgiTL,andadd f´c&fe+LC hv(gi*)%k#t^07xtoUUqiTL.)3)7xRepGeatstep2)asoftenaspossible.When nallytheintersection7xSuppMp(m)8\(L*T %(giTL))UUisUUempty*,increaseibyone.)4)7xIfIIis,IIcontinuewithstep2).OtherwisesetIIp=mandIIreturnthe7xlistUU[q1|s;:::;qsF:;p].%ShowGthatthisisanalgorithm,i.e.thatitstopsafter nitelymanysteps,%and thatitreturnsalist [[q1|s;:::;qsF:];p]such thatq1|s;:::;qs R2Pc,p2Pc^r1,%andUUm=q1|sg1S+8g+8qsF:gs+p.?d)%Giveanexampleinwhichtherepresentationcalculatedinc)doGesnot%haveUUthepropGertiesrequiredinTheorem1.6.4.\Ie)%ShowF/that,ifonerepGeatsthealgorithmofc)oftenenough(i.e.ifone%applies@ittotheelement@pinstead@ofm,etc.),therepresentationsofm%one=getsbGecomeeventually=stable.Giveanexampleinwhichthisstable%representationf+stilldoGesnotagreewiththerepresentationcalculatedby%theUUDivisionAlgorithm.K~Wō>;V1.6TheTDivisionAlgorithmٮ75-ō>;f)%ImplementS%thealgorithmofc)andtheproGceduredescribedine)ina %C0oLCoACfunctionDivision23(::: )andcompareitseciencywiththefunc-%tionUUDivision.U=(::: )UUbyUUapplyingittothetestcasesofb). T utorialT15:NormalRemainders궲IfQweareonlyinterestedinthenormalremainderofanelementQm2Pc^r withrespGecttoatupleofvectorsGѲ,wecanuseasimpli edversionoftheDivisionAlgorithmUUwhichwewanttoexamineinthistutorial."Let>KAZbGe>a eld,P-~=K[x1|s;:::;xnq~]apGolynomialringoverK,amoGduletermorderingonT^nq~he1|s;:::;ermi,andG`=0(g1|s;:::;gsF:)2(Pc^r1)^sF:.T*oeachUUvectorUUm2Pc^r1,wecanapplytheNormalTRemainderAlgorithm.1)%ChoGosethelargesttermt2Supp(m)withrespGecttowhichisdivisible%byRoneoftheleadingtermsRL*Ttw(g1|s);:::;L*TN7Ͳ(gsF:).Ifnosuchtermexists,%returnUUmandUUstop.2)%FindetheminimaleiYݸ2f1;:::;sgsuchethatL*TP(giTL)dividestandwrite%t=t^0xL*Tp(giTL)UUwitht^0Q2T^nq~. ;3)%Let~c2KOn3f0gbGe~thecoecientof~tinm.~Replacembym3 'ctr0Jf&fe+LC hv(gi*) 7ĵgi ˍ%andUUcontinuewithstep1)."Asweshallsee,forthepurpGosesofSection2.5,itwillsucetoimplementandUUusethisalgorithm.a)%ProvethattheNormalRemainderAlgorithmisanalgorithm,i.e.that%it<stopsafter nitelymanysteps.ThencompareittotheDivisionAlgo-%rithmUUandshowthatitreturnsUUNR1ȟG۲(m).?b)%W*riteYaC0oLCoA!programYNormalRemainderR,(::: )Ywhichcomputesthenor-%mal}remainderofanelement}m S2Pc^r NCwith}respGecttothelistofvectors%G jusing jtheabGove jalgorithm.Donotusethebuilt-infunctionNRd(::: )of%C0oLCoA@֖.\Ic)%ApplyLtheprogramLNormalRemainderS (::: )Linthefollowingcases,where%K~4=QUUand"=PosLex#G.)1)8xmf=x^4l1|sx2[+:x^4l2x3+:x^4l3x1l2fQ[x1;x2;x3],Nݵg1l=fx^2l1x2|s,g2l=fx^2l2x3|s,7xg3C=x^2l3|sx1)2)8xmr=(x^3l1+J 1;x^3l2+1;x^3l3+1)r2Q[x1|s;x2;x3]^3|s,g1 x=r(x1;x2;x3),7xg2C=(0;x2|s;x1))3)8xm=(x1|sx20+x3x4;x1x2x3x4)2Q[x1|s;x2;x3;x4]^2|s,õg1C=(x1;0),õg2C=7x(x2|s;0),UUg3C=(0;x3),UUg4C=(0;x4)?d)%Give(anexamplewhichshowsthatthenormalremainderofanelement%m2Pc^r &depGendsUUontheorderingoftheelementsinUUG^=(g1|s;:::;gsF:).\Ie)%GiveCanexampleofanelementCmK2hg1|s;:::;gsF:iPc^r tsuchCthat%NR4G:E%(m)6=0.nShowthatifanelementnm2Pc^r ?ֲdoGesnsatisfyNRKG+(m)=0,%thenUUitiscontainedinthesubmoGduleUUhg1|s;:::;gsF:iPc^r1.L܀Wō>;764`1.F:oundations-ō>;1.7**GradingsTheN; E1.7Gradingsٮ77-ō>;\Ic)%IfyĵrJ2еRandr=ПP  n92r RisythedecompGositionofyĵraccordingtoa.1), %where_r v2R !,_thenr .iscalledthehomogeneouscompQonen9tof%degreeUU ㍲ofrG."IfD RWҲisD ac-gradedring,then0isahomogeneouselementofRWҲofeverydegree.Moreover,thedecompGositionofeveryelementintoitshomogeneouscompGonentsoisunique,sinceinDe nition1.7.1.awehaveadirectsum.Ifthecancellation#lawholdsin#c,thenthesetR0visasubringofRDz,andforevery궵 UP2theUUsetUUR visanR0|s-moGdule."ThefollowingtwoexamplesconstitutethemostimpGortantsituationsinwhichUUweshallmeetUUc-gradedrings.ExampleT1.7.2.c;784`1.F:oundations-ō>;ExampleT1.7.5.c=gS[x1|s;:::;xnq~] bGeiapolynomialringoveriSyequippediwiththeiN^nq~-gradingde nedinEx-ample1.7.3.ViatheisomorphismlogX:T^n {'!N^nq~,weshallviewthisasaT^nq~-grading.[XF*or[Xr:1,thesetoftermsT^nq~he1|s;:::;ermiofthePc-moGdulePc^r궲isvBavBT^nq~-monomoGdule.NowweletvB(Pc^r1)tei y=Steiʎfort2T^n and1irG,i.e."ufor"uteiqB2T^nq~he1|s;:::;ermi.Itiseasytocheck"uthatthismakesPc^r into"uaT^nq~he1|s;:::;ermi-gradedUUPc-moGdule.󎍑"GivenEaE-gradedRDz-moGdule,thereexistsacheapwayofmakingmore궵-gradedA.RDz-moGdulesA.calledshiftinglde}'grees.A.ModulesobtainedbyshiftingdegreesسinسRzitselfarethebasicbuildingbloGcksintheconstructionofgradedfreeUUresolutionsinV*olume2.De nitionT1.7.6.is~LetB UP2ѲbGeBa xedelementsuchthatthemultiplicationmap+ 覲:Z w!de ned+bys7! ss;sis+injective.F*orinstance,iftheleft-cancellationUUlawholdsinUU,thisassumptionissatis edforall UP2c.a)%F*oreveryϵs2,wede neϵM( 8)s =M n9s.ThenweletϵM( 8)=%s2@M( 8)sF:.Itiseasytocheckthatinthiswaywegeta-graded%RDz-moGdule M( 8). W*ecallitthemoduleobtainedbyshiftingEdegrees%by# 8.#Ifthemap A: ω!is#bijective,thesetunderlyingM( 8)%agreesUUwithUUM.?b)%MoGdulesoftheformi2I pRDz( iTL),whereIRisasetand id2Kforalli2I,%willdbGecalleddc-gradedBfreeRDz-modules.dHereweletd(i2I pRDz( iTL)) =%i2I pRDz( iTL) vforUUallUU UP2c."HavingEde nedEc-gradedringsand-gradedRDz-moGdules,wealsoneedtheUUappropriatesetsofhomomorphismsbGetweenUUthoseob8jects.De nitionT1.7.7.is~LetS7bGeanotherringwhichisgradedoveramonoid(c^01ȵ;+),UUandletUUNlpbGeanother-gradedRDz-moGdule.a)%F*ormaringhomomorphismm':Rݸ!S{andmahomomorphismofmonoids% ":*!c^01Ȳ,^wecall^('; [ٲ)(or^simply')ahomomorphism5ofgraded%ringsUUifUU'(R !)S: @L( n9)A޲foreveryUU UP2c.?b)%An`RDz-linear`map:M3!Nwiscalledahomomorphismof-graded%R-moQdules+orahomogeneous }R-linear }mapif+(MsF:)|Nsefor+all%s2."F*or$instance,let$ 2pƵbGeaninvertible$element,andlet$r2pƵR !.ThenthehǵRDz-linearhmaprU":瀵RDz( 8)!R|de nedbyhǵrG^0ָ7!rGr^0~isahomomorphismofUUgradedUURDz-moGdules."NextXvwewanttointroGducethe\correct"kindofsubob8jectsofgradedringsandѸmoGdules.NotethatifweequipѸP*=K[x]withѸthestandardgradingandlet)޵I2=)P(xƐ1)Pc,)thenitisclearthat)޵Ir\ƐPd=(0)for)everyd2N.SomehowJ4thissuggeststhatJ4IdoGesnot\inherit"thegradingofPc.Inotherwords,whatwereallyneedisthatthecanonicalinjectivemapIuF,UXq!dPByisamhomomorphismofgradedmRDz-moGdules.Spellingthisoutinconcreteterms,weUUarriveatthefollowingde nition.OWō>; E1.7Gradingsٮ79-ō>;De nitionT1.7.8.is~An2RDz-submoGduleNJ of2the-gradedR-moGduleMJ is calledUUaUU-gradedTR-submoQduleofMlpifwehaveUUN3=s2@(NO\8MsF:)."Ac-graded2submoGduleof2Risalsocalledac-homogeneous7^ideal궲ofHRDz,HorsimplyahomogeneousƟidealofR\Rifisclearfromthecontext.kRemarkT1.7.9._uHLetNM .bGea-gradedRDz-submodule.W*ecanequiptheOresidueclassmoGduleOM=qNg withthestructureofa-gradedRDz-moGdulebyde ning(M=qN)s0=MsF:=Ns forevery۵s2.Thusthecanonicalhomo-morphismUM3 !M=qNpbGecomesUahomomorphismof-gradedRDz-moGdules.Inuparticular,theresidueclassringuR=I>DzofRbyuahomogeneousidealisagainUUaUUc-gradedring."F*orOpracticalpurpGoses,thefollowingpropositionanditscorollaryaremostuseful.BTheyallowustoquicklyprovethatsomesubmoGduleisBԵ-gradedbyexhibitingahomogeneoussystemofgenerators,andtousethisfacttoget\nice"5representationsofarbitraryhomogeneouselementsintermsofthosehomogeneousUUgenerators.PropQositionT1.7.10.wL}'etڈN3MbeڈanR-submodule,ڈandletNs R=N\Msforalls2.Thenthefollowingc}'onditionsareequivalent.~oa)&N3=s2@NsHb)%Ifn2N؛andn=P s2"ns isthede}'compositionofnintoitshomo}'ge-%ne}'ouscomponents,thenns R2Nforalls2.Hc)%Ther}'eVisasystemofgeneratorsofVNqwhichconsistsofhomogeneous%elements.Pr}'oof.6First,weshow,a)&)b).,ChoGoseanelement,n&2NGand,letn=궟Pxs20d%ns9bGeitsdecompositionaccordingtoa),wherens R2Ns9foralls2.Since/۵ns R2NsMsvand/۵M3=s2@MsF:,/thisisalsothedecompGositionof/۵n궲intoitshomogeneouscompGonentsinM.ThusthehomogeneouscompGonentsofUUnlieUUinN."Implication`b))c)follows`bytakingallhomogeneouscompGonentsofasystemofgeneratorsofN.Nowweshowc)!)a).Letfn j ?=2Bqg궲bGeƖahomogeneoussystemofgeneratorsofƖNݱandletn.2N.ƖW*ewrite궵n犲=Puş \2B"Jr dn withelementsr Lp2RDz.F*oreach n2BrwedecompGose궵r +=P US n92s۵r \; ɲintoUUitshomogeneouscompGonents.ThenH2/d{n=tX  \2BMX 7A n92&ɵr \; ftn +=nzX 7s22L`;X jطf( \; n9)]j deg (n ʱ)=sgklr \; n d`$2nzX 7s22LNs^궲showsN =P-s2!oaNsF:,andfromM =s2@Ms wegetthatthissumisdirect.(PSCorollaryT1.7.11.lSupp}'ose thattheright-cancellationlawholdsin .Let궵N Mb}'eʚaʚ-gradedR-submodule,ʚletfn ^׸j  2Bqgb}'eʚasetofho-mo}'geneousG/generatorsofG/N,andlets ߸2.EveryelementG/n2Ns ihasar}'epresentationKnz=PV \2B"+޵r dn 1withKhomo}'geneouselementsKr -`2zRsuchthatdegv(r d)8deg\o(n )=sforevery N42Bq.P"Wō>;804`1.F:oundations-ō>;Pr}'oof.6Let{tn=P ܟ \2B ja dn Zwitha k2R;for 2Bq.{tW*edecompGosea Zas the sumofitshomogeneouscompGonentsandgroupthembywriting a Z=궵a^0v +a^00v d,wherea^0v JistheuniquehomogeneouscompGonentofa Jsuchthatdeg#E(a^0v d)ĸdegS(n )=s.W*egetǵn=P US \2B *|a^0v n .+ğP W \2B-(a^00v n d.Equivqalently*,wehhaveh0p=(P ; \2Bcda^0v dn Eеn)+P  \2B4a^00v n d.hByconstruction,theelement궟Px \2B1Na^0v dn ynbisbahomogeneouscompGonentofthissum,i.e.ithastobezero./7="궲IntheprecedingproGofweusedtheassumptionthattheright-cancellationlawholdsin1inordertohavea^0v uniquelysingledoutbytherelationdeg#E(a^0v d)cиdeg_(n )2q=s.W*eleavethegeneralizationtothereader(seeExer-ciseUU9).)"F*ortheremainderofthissection,weshallassumethatwearealsogivenamonoidorderingȲonc.ByRemark1.4.2.b,thisimpliesthatthemonoid궲is)in nite.W*ecancharacterizehomogeneousprimeidealsbytheusualprop-ertyUUappliedtohomogeneouselementsonly*.fPropQositionT1.7.12.wL}'etRpbeRahomogeneousproperidealinR.Thenthefollowingc}'onditionsareequivalent.~oa)%Theide}'alpisaprimeide}'al.Hb)%Iffg"2pforhomo}'geneouselementsfV;g"2R,thenfڧ2porg"2p.Pr}'oof.6Itsucestoshowthatb)impliesa).LetfV;gH2FoRbGetwoelementssuch@cthat@cfg"2p.W*edecompGosethemintotheirhomogeneouscompGonents.If weallowsomecompGonentstobGezero,wemayassumethatthetwosetsofdegreesareidentical,i.e.thatwehaveڵfmֲ=ZGf 1 +@++@f smandg =궵g 1 +B/Ų+Bg s ,cwherec۵ 1[<ނhv<ނ sF:.F*oracontradiction,weassumethatthe~numbGers~i =minofk\X2Njf k := 52ԢpgandjL=minofk\X2Njg k := 52Ԣpgexist.Now0weloGokatthehomogeneouscomponentofdegree0 i+~ǵ jܲoffg[ٲ.ItisgivenUUbytheformulaJ1syf iMg j + 4X j8f(k+B;l `)j k_+ lұ= i*+ jagOwf k tg l!-궲Since3& k $X< 7 iror lJ< j iҲfor3&everysummandabGove,wehave3&k_<8ior궵lw{<Ej,:andtherefore:f k tg l B2p.Hencealso:f iMg j AbGelongstop,andthehypGothesisUUimpliesUUf i e2porg j 6θ2p,UUacontradiction.V#="궲AlthoughLIneitherofthetwoLIequivqalentconditionsinthepreviouspropGo-sition]containsanyreferenceto]!Dz,theexistenceofsuchamonoidorderingisinstrumentalUUfortheclaimtohold,asournextexampleshows.ExampleT1.7.13.h`Let'R= Z[i]bGe'theringofGauiannumbers'(seeT*u-torialV4),i.e.theVZ-subalgebraofCgeneratedbyfig.Ifweusethegroup궵ò=4Z=(2),fweseethatfR-isac-gradedringwithR0 =4ZandR1=4Zi.LetW?usconsidertheidealW?I|=(2)inRDz.W?Itisnotaprimeideal,sinceQ5SWō>; E1.7Gradingsٮ81-ō>;궲(1uϸi)(1+i)=2andneither1uϸi̲nor1u+i̲isinI.However,if̵fV;g"2Rare homogeneous[elementssuchthat[fg-2I,thenf夸2I$ϲorg-2I.[Thisfollows,bGecausegeithergfV;g@d2䋵R0torf;g@d2䋵R1|s,gandinbGothcasesthefactthat2isaprimeUUnumbGershowstheclaim."OurlastgoalinthissectionistoproveaversionofNakqayama'sfamouslemmaUUadaptedtogradedmoGdules.First,weneedthefollowingresult.LemmaT1.7.14.bG If&@is&atermor}'deringonc,thenR+ f=X n9>|0R @isahomo}'geneousidealofR.Pr}'oof.6ItzsucestoshowzRȸR+ RzR+.Thisfollowsfromthefactthatevery_elementisa nitesumofhomogeneouselements,andforhomogeneouselementsvf2qR !,gZJ2R n90 Lwith 8; ^0Z2andv ^0Z>0wevhavefgZJ2qR n9+ 0궲withUU +8 8^0#>M UP0byUUDe nition1.4.1._PropQositionT1.7.15.w(GradedTV ersionofNak\ra9yama'sTLemma)L}'et1beatermor}'deringonsyandkawell-or}'deringonwwhichisc}'ompat-iblewith!.Supp}'osethattheright-cancellationlawholdsin.LetM1|s;M2b}'es%twos%-gradedR-submoduless%ofM@suchthatM1CM2M1n+R+9M2|s.ThenM1C=M2|s.Pr}'oof.6ItsucestoshowM2 &"M1|s.SuppGosethatthisisnotthecase.ByLPropGosition1.4.18,thereexistsahomogeneouselementLmcq2M2ZCnеM1궲of;minimumdegreewithrespGectto;[ٲ.UsingthehypGothesisandCorol-lary1.7.11,weseethatthereexisthomogeneouselementsm^0 72iVM1|s,궵g1|s;:::;gsS2 }M2|s,,and,f1;:::;fsS2 }R+ such,that,m=m^0+oPލ Is% Ii=1hfiTLgimxandsuchthatdeg,x(fiTL) degØ(gi)=deg(m)fori=1;:::;s.Sincedeg,x(fi)>M0for궵iwӲ=1;:::;s,_thedegreesoftheelements_g1|s;:::;gs arelessthanthedegreeofm.Thechoiceofmthenimpliesg1|s;:::;gsJ2^M1|s.Consequently*,weget궵m2M1|s,UUacontradiction.-CorollaryT1.7.16.lL}'etFhybeFatermor}'deringonAandawell-or}'deringonhQwhichhisc}'ompatiblehwith!.Supp}'osethattheright-cancellationlawholdsin.~oa)%A setofhomo}'geneouselementsm1|s;:::;ms$2IM9generatestheR-mod-%uleG%M^@ifG%andonlyiftheirr}'esidueclasses~feǷgmܟ1O;:::;~feǷgm r_sinM=(R+ ^RµM)%gener}'atethisresidueclassmodule.Hb)%IfzR0 isza eld,everyhomo}'geneouszsystemofgener}'atorsofMc}'ontains%aminimalone.Pr}'oof.6T*oprovea),itsucestoshowtheimplication\(".LetNƲbGethegradedosubmoGduleofoMgeneratedbyfm1|s;:::;msF:g.Byassumptionwehave궵M3NO+8R+pM.UUThereforeNakqayama'sUULemmayieldsUUM=N."TheLproGofofb)followsfroma),fromLR=R+T͍ +3 =0ĵR0|s,andfromthefactthateverysystemofgeneratorsoftheR0|s-moGduleM=(R+ 2:M)containsabasis.,RHWō>;824`1.F:oundations-ō>;%Exercised1.Let:bAe:amonoidinwhic9hthecancellationlawholds,and %letTR%bAeTaH-gradedring.Pro9vethatT12R0*.%Hint:TW:riteT1=?P UP c2yr Uandsho9wthatTr0m=1.u%Exercise2.hLet1:=Wf0;1gbAethemonoidde nedinExercise2of%SectionT1.3,andletTR%bAearing. *?a)7xDe nedegc(rA)H=1forallr2HR>.Sho9wthatthismakesR(intoaH-7xgraded-ringinwhic9htheelement1ishomogeneousofsomenon-zero7xdegree.)b)7xEquipS=[Rn^RZwithcompAonen9twiseadditionandmultiplication,7xand9let9S0m=Rg-V0asw9ellasS1 g=0VR>.Sho9wthatthismakes9Sinto7xaH-gradedringinwhic9htheelement1=(1;1)isnothomogeneous.%Exercisen3.LetR-bAearing,letP4=R>[x1*;::: ;xn7],andletr-1.%Chec9kMthatthede nitionMPt6=RHg8)tfort2T-=nȄmak9esP0intoMaT-=n7-graded%ring,ޱandthatޱPH-=r Ttogetherwiththegradingde nedinExample1.7.5isa%T-=n7he1*;::: ;er,pi-gradedTPH-moAdule.%Exercise4. pyLet)K-bAe)a eld,letR =~K[x],letFa=~N,andlet%R ^=fc(x81)-= jc2KgTforTall 2H. *?a)7xSho9wTthatTR%isaH-gradedring.)b)7xPro9veTthatTIF!=(x)isnotahomogeneousidealofR>.*6c)7xGiv9eTanexampleofahomogeneousidealofTR>.%Exercise5.Chec9kthatthesetM( )<=s2 M( )s,asin9troAduced%inJDe nition1.7.6,isindeedaJ2-gradedR>-moAdule.JThenconsideran%elemen9tFr@u2R GandFthemapM !M( )de nedb9ym7!rAm,Fand%sho9wTthatitisahomomorphismofT2-gradedR>-moAdules.%ExerciseG6.*LetqMK5bAeqMa eld,letR;N=+K[x1*;x2],qMlets=+N-=2*,andlet%Rfh( 'Ӱ1 ; 'Ӱ2)A*==Afcx 'Ӱ1Ѝ1 :x 'Ӱ2Ѝ2 xjc2Kgfor( 1*; 2)=A2N-=2 @xasinExample1.7.3. %F:urthermore,letH-=0p=NandletRfh cG0 obAetheK>-v9ectorspacegeneratedby%fx 'Ӱ1Ѝ1 :x 'Ӱ2Ѝ2 j 1Z+/l 2o= -=05UgG"for -=02H-=0.G"Inthisw9ay:,G"RW`bAecomesG"bothaG"H-%andsasH-=0-gradedring.Finally:,let':R8!R߱bAestheiden9titysmapand% _:!H-=0!the'Jmapde nedb9y'J R(( 1*; 2))= 1D+1 2*.'JShowthat'J('; R)%isTahomomorphismofgradedrings.%Exercisek57.תLetR bAeaH-gradedringandSݸZR asubring.Discuss%whethertandho9wonecanequiptSwithaH-gradinginsuc9hawaythat%theTinclusionTS,uV!R%bAecomesahomomorphismofgradedrings.%Exercisez8.oLet`bAeamonoidinwhic9hthecancellationlawholds.A%H-gradedQringQR= c2:R RiscalledaH-gradedLL eldifev9eryhomoge-%neousTelemen9tofTRvn8f0gisTaunit.LetTR%bAeaH-graded eld. *?a)7xPro9ve,that,R0 Aisa eld,andthatforev9ery 2?`theR0*-vector7xspaceTR UhasTdimension1.(Hint:UseExercise1.))b)7xGiv9eanexampleofaH-graded eldwhichisnota eld.(Hint:Con-7xsiderTtheringTK[x;x-=1 ].)*6c)7xSho9wTthatTf 2کjR ^6=0gisTagroup.)d)7xLet٭MґbAe٭a nitelygeneratedH-gradedR>-moAdule.Pro9ve٭thatM7xhasanR>-basisconsistingofhomogeneouselemen9ts.(Hint:Startwith7xaRminimalhomogeneoussystemofgeneratorsandsho9wthatitisa7xbasis.)S\Wō>; E1.7Gradingsٮ83-ō>;%Exerciseg9.?MoAdifyHthestatemen9tofCorollary1.7.11togeneralizeitto %the`caseofamonomoAdule`|inwhic9htheright-cancellationlawdoAesnot%necessarilyThold.%ExerciseW10.@Giv9e9anexampleofaring9Rwgradedoveramonoid9(A;+)%suc9hthatthereexistsamonoidordernonH,butR+4= c>W0R isnot%anTidealofTR>.=T utorialT16:HomogeneousP9olynomials궲RecallthatthestandardgradingonthepGolynomialringPt\=͵K[x1|s;:::;xnq~] overWa eldWյKwasde nedbyWյPdt=BffѸ2P.Ѹjdeg(t)=dUUforall t2Supp*(f)g궲forB$d2NinB$Example1.7.2,andthattheelementsofPdֲarecalledhomoge-neousUUpGolynomialsofdegreeUUd."In3thistutorialwewanttogetabGetterunderstandingofthespace3PdݧofhomogeneouspGolynomialsofdegree˵d.W*ewanttoknowitsdimensionandtoUUcharacterizeitselements.a)%ShowUUthatUUPdisaK-vectorspaceofdimensionb ꬴn+d1⾍md"eb+Pforalld2N.?b)%FindIandproveIaformulawhichcomputesthedimensionofthequotient%vectorspaceofҵPd=V8,whereVisthesubspaceofpGolynomialswhichare%divisibleUUbyUUx1|s.\Ic)%LetK>bGeanin nite eld.Supposethatfڧ2P"satis esf(a1|s;:::;anq~)=0%forUUeveryUU(a1|s;:::;anq~)2K^n(.ThenshowthatUUfڧ=0.?d)%ProGduceOanexamplewhichshowsthattheabGovestatementisfalseifOK%isUU nite.\Ie)%Prove\thatif\f2WP=isanon-zerohomogeneouspGolynomialofdegreed,%then9ȵf(a1|s;:::;anq~)C=^dz޸,f(a1|s;:::;anq~)9holds9forallCָ2Kand9all%(a1|s;:::;anq~)2K^n(.f)%Provethattheconverseofe)holdsifKӿhasatleastmaxfdeg#(f);dg}+1%elements."Nowconsiderthefollowingnon-standardgradingonPM=-K[x1|s;x2].W*edeclare`'x1 ܚto`'bGehomogeneousofdegree2andx2 ܚtobGehomogeneousofdegree#%1.#%ThenweletPd rw=ŸffT2P,Tj2 12+l 2 E8=dUUforall x: 1l1 x: 2l2bǸ2궲Supp*#(f)g8[f0gUUforUUalld2N.g)%ProveUUthisde nitionmakesUUPintoUUaN-gradedring.?h)%ExplicitlyQjdescribGethefunctionfromQjNtoNwhichQjmapsdtodimK˲(Pd),%i.e.UU ndaformulaforUUdimK(Pd).i)%MoGdifyUUe)aboveUUto tthiscase.TpWō>;-ŎUMLe @궄jbRō>;-ō>;궾2.%NGrgobnerffBasesvėWANIfwehaveabunchofnon-zerovectors>fg1|s;:::;gsF:g,wegetabunchofrewriteUUrules.Whatkindofgamecanweplaywiththoserules?"SuppGosecavectorcm˸2Pc^r 4containscaterminitssuppGortwhichisamul-tipleofL*TZ(giTL)forsomei2f1;:::;sg.ThenwecanusetheruleassoGciatedtogiCвandrewritem.Theelementobtainedinthiswayiscongruenttom궲moGdulo_M._Theprocedureofmovingfromonerepresentativeofthisresidueclass!toanotherresemblesthedivisionalgorithm.However,ateachpGointwemay!Yhaveseveralmovesavqailable,andadi erentorderofthosemovescouldlead1>toadi erentresult.A1generatingset1>fg1|s;:::;gsF:gofMHYis1>spGecialif,nomatterwhichorderyouchoGose,youalwaysarriveatthesameresult.InSectionܙ2.2,wetreatrewriterulesandprovethesurprisingfactthatthisnewkindUUofspGecialtyisequivqalenttotheonesdescribGedbefore."However,themostfundamentalmotiveforloGokingatspecialsystemsofgeneratorsܶisstillmissing.Thenotionofasyzygyofatupleܶ(g1|s;:::;gsF:)isVWō>;864`2.Gr`obnerTBases-ō>;궲oneofthedecisiveideasforsuccessfulapplicationsofComputationalCom- mutative@-Algebra.UsingthetheoryofgradingsdevelopGedinSection1.7,weshowthateverysyzygyof(L*T %(g1|s);:::;L*TN7Ͳ(gsF:))canbGeliftedtoasyzygyofP(g1|s;:::;gsF:)ifPandonlyifPfg1;:::;gsF:ghasPthespGecialpropertiesdiscussedearlier."After$threateningtodoitforalongtime,we nallycombineallthoseideasYandintroGduceGrobnerbases.A%GrobnerbasisofasubmoGduleYMtofPc^r궲ishmasetofgeneratorswhichisspGecialinone(andthereforeall)oftheaboveways.ߒInSection2.4welaunchaninvestigationintotheirpropGertiesandusesby5showingthattheirexistencecanbGeviewedasaconsequenceofDickson'sLemma.۸MostapplicationsofGrobnerbaseswillbGetreatedinChapter3andV*olume*2,butsomerewardsforourcarefulpreparationscanbGereapedimme-diately*,bforinstanceaproGofofHilbert'sBasisTheorem,thenotionofnormalforms,|thesubmoGdulemembershiptest,andanewversionofMacaulay'sBasisUUTheorem."NextweputagreatemphasisonthederivednotionofareducedGrobnerbasis.IthastheastonishingpropGertythat,givenasubmoGduleMofPc^r궲andatermorderingϵ[ٲ,itisauniquesystemofgeneratorsofMsatisfyingcertainnaturalconditions.W*ebGelievethatthisisoneofthemostubiquitoustheoretical%ZtoGolsinComputationalCommutative%ZAlgebra.Justtogivethe avour9ofitsimpGortance,weshowhowonecanuseittodeduceaseeminglyunrelatedK+resultabGouttheexistenceanduniquenessofthe eldofde nitionofUUsubmoGdulesofUUPc^r1."Afterqallthistheory*,itistimetoexplainhowonecanactuallystepintoactionandcomputeaGrobnerbasisofMҭfromagiven nitesetofgener-ators.g@ThepGowerg@ofourstudyofsyzygiesenablesustocapturethespiritofBuchbGerger's50AlgorithminSection2.5.Notonlyshallweproveandimproveits basicproGcedure,butweshallalso nallyachieveourgoalofe ectivelycomputingseinresidueclassmoGdulesviaMacaulay'sBasisTheoremandnor-malUUforms."AssometimeshappGensinreallife,includingscience,thediscoveryofatoGol7fwhichenablesustosolveoneproblemopGensthedoortomanyotherdiscoveries.GrobnerbasesarecertainlyoneofthosetoGols,butbeforedelvingintotherealmoftheirapplications,weclosethechapterwithanotherone,namely~HilbGert'sNullstellensatz.ThistheoremisoneofthemilestonesintheproGcessoftranslatingalgebraintogeometryandgeometryintoalgebraandCformsthebackgroundformanyapplicationsinalgebraicgeometry*.Sec-tionYH2.6isentirelydevotedtoitsproGof,whichalsousessomepiecesofGrobnerbasis6{theory*.IthighlightstheimpGortanceofswitchingfromoneground eldtoMXa eldextension,sothatthegeometricnotionofananevqarietygetsitspropGerUUperspective."Once\morethechaptercloseswithanop}'eningtheme.BesidesbGeingametaphorOoflife,thisendofonestrugglealreadylaysthegroundworkforsuccessfulUUapplicationsinsubsequentchapters.WWō>;|2.1SpAecialTGenerationٮ87-ō>;2.1**Sp`ecialGeneration.ލAÎlxlNy[ٲ(x^251)x(xy`1)=xy,+andR=DegLex*theleading3monomialsofthetwo3summandscancelout,sothat3x,theleadingtermyoftheresult,issmallerthantheleadingtermsofthesummands.Thisshows=thatsomegeneratorshaveaspGecialbehaviourwithrespecttotheleadingtermsoftheelementstheygenerate.Moreprecisely*,weseethat궵xɲ=L*TX(f);=2 >(L*T %(x^2͸JZ1);L*TN7Ͳ(xy31))=(x^2|s;xy[ٲ).oHowever,ifweaddinthiskexampletheelementsguaranteedbyPropGosition1.5.6.b,wegetanothersetofgeneratorsoftheideal(x^2pd1;xyָ1)whoseleadingtermsgeneratetheUUleadingtermideal."Thisistheprototypicalcaseofthephenomenonthatnotallsystemsofgener}'atorsGofanidealormoduleareequalalludedtointheintroGductionofthis.lchapter.SomesystemsofgeneratorshavespGecialpropertieswhichwewantftodescribGeinthisandthefollowingsections.Lateritwillbecomeclearthat!YallofthosepropGertiesareincarnationsofthesameconcept,namelytheconceptUUofGrobnerbases.o"AsNusual,weletNKjbGea eld,n 1,P=K[x1|s;:::;xnq~]aNpGolynomialring,UUr51,UUand.amoGduletermorderingonT^nq~he1|s;:::;ermi.PropQositionT2.1.1.q(SpQecialTGenerationofSubmodules)L}'etwMZ?Pc^r cbewaPc-submodule,wandletg1|s;:::;gsy2?Pc^r‡nVf0g.Thenthefollowingc}'onditionsareequivalent./@A1|s)*CF;oreveryelementm}ڸ2M|naf0g,ther}'earef1|s;:::;fs2}ڵPsuchthat)Am=Pލ USs% USi=1tJfiTLgiNandUL*T['(m))L*T=/Ӳ(figi)UforUalli=1;:::;sUsuchthat)AfiTLgid6=0./@A2|s)*CF;or5everyelement5m2Mnhf0g,5ther}'eare5f1|s;:::;fs R2P[suchthatm=)APލ3޴s%3i=1BյfiTLgi3andL*T7v (m)=maxc$fL*T %(figi)ji2f1;:::;sg;figid6=0g.Pr}'oof.6SinceGConditionGA2|s)obviouslyimpliesA1|s),itsucestoprovethereversedirection.Theinequality\b"inµA2|s)followsimmediatelyfromµA1|s).TheUUinequality\b"inUUA2|s)followsUUfromPropGosition1.5.3.a.6b&"궲If^MM2Pc^r /ʲis^aPc-submoGduleandg1|s;:::;gsl22MҸnf0g,thenCondi-tionsBA1|s)andA2)sayBthatfg1;:::;gsF:gisBaspGecialsystemofgeneratorsof(M.(UsingtheexamplementionedabGove,weseethatitisnottruethatXܠWō>;884`2.Gr`obnerTBases-ō>;궲ConditionsK A1|s)andA2)holdK foreverysystemofgeneratorsofK M,bGecause L*T E%۲(f)< max!fx^2|s;xy[ٸg maxfL*T %(f1|s(x^2 v1));L*TN7Ͳ(f2(xyO1))g,طin-depGendentUUofwhichelementsUUf1|s;f2C2Pon8f0gweUUchoGose."ItisalsointerestingtoobservethatifGisatermorderingonT^n andYisaemoGduletermorderingoneT^nq~he1|s;:::;ermiwhicheiscompatiblewithe!Dz,thenweUUcanexpandUUL*T[z(fiTLgi)=L*Tjiܲ(fiTL)L*TN7Ͳ(gi)UUinUUtheabGovestatements."TheintuitivemeaningofConditionsA1|s)andA2)isthateveryelement궵mx߸2M~ncf0gYshouldYhavearepresentationYmx߲=Pލs%i=1&fiTLgisuchthatthehighestMtermwhichoGccursinthecomputationoftheright-handsidedoGesnotJcancel.Consequently*,theleadingtermofJmisJamultipleofoneofthetermsL*T6V(g1|s);:::;L*TN7Ͳ(gsF:).NowweexaminethislastpropGertymoreclosely*.PropQositionT2.1.2.q(GenerationTofLeadingT ermMoQdules)L}'etM3Pc^r beaPc-submoduleandg1|s;:::;gs R2MKn3f0g.Thenthefollowingc}'onditionsareequivalent./@B1|s)*% The8set8fL*T %(g1|s);:::;L*TN7Ͳ(gsF:)ggener}'atestheT^n-monomo}'duleL*T1ǟ]fMgUX./@B2|s)*% TheҢsetҢfL*T %(g1|s);:::;L*TN7Ͳ(gsF:)ggener}'atesthePc-submo}'duleL*Tv1Dz(M))AofP^rm.Pr}'oof.6SinceޞB1|s)impliesB2)byޞde nition,itsucestoshowthereversedirection.Letm2Mnf0g,andletL*Tr(m)=f1'L*Tʪ-@(g1|s)+ +fsL*Tq(gsF:)forsomepGolynomialsԵf1|s;:::;fs ˸2Pc.ByProposition1.5.3.a,thetermL*T E%۲(m)\is\inthesuppGortofoneofthevectorsf1'L*Tʪ-@(g1|s);:::;fsL*Tq(gsF:).Thus@+thereisanindex@+iN{2f1;:::;sgand@+atermtN{2Suppc(fiTL)such@+thatL*T E%۲(m)=t8L*To?(giTL).A2׍"궲Finally*, weshowthe rstimpGortantlinkbGetweenthetwopropGertiesofspGecialUUsystemsofgeneratorswhichwehavedescribGedsofar.PropQositionT2.1.3.qL}'etbMbGPc^r bebaPc-submodule,bandletg1|s;:::;gs ]b}'enon-zer}'oelementsofM.ThenConditionsA1|s),A2)ofPr}'oposition2.1.1andConditionsB1|s),B2)ofPr}'oposition2.1.2areequivalent.Pr}'oof.6ConditionIA2|s)impliesB1)byIPropGosition1.5.3.d.Thusweshow궵B1|s)d)A1).MSuppGosethereexistsanelementMmd2Mnff0gwhichMcannotbGe‰representedinthedesiredway*.ByTheorem1.4.19,thereexistssuchanelementcmwithcminimalleadingtermwithrespGecttoc[ٲ.ByB1|s),wehaveL*T E%۲(m)}=t\<L*T˟ba(giTL)_for_somei}2f1;:::;sg_andsomet}2T^nq~.Clearly*,we havemR&hLCW(m)ʉfetLC k(gi*)!˵tgid6=0,sincem=&hBLC^(m)BʉfetLC k(gi*)#7.tginͲwouldbGearepresentationLcsatisfyingRA1|s).RThereforewe ndL*T@Xֲ(m3&hf˱LC A(m)f˟ʉfetLC k(gi*)![tgiTL)<)L*T=/Ӳ(m),andtheelementPmظ&hN LC *(m)N ʉfetLC k(gi*) BQtgid2M1nf0gcanPbGerepresentedasrequiredinA1|s).But ˍthen_also_mcanbGerepresentedasrequiredinA1|s),incontradictionwithour assumption.1YLWō>;|2.1SpAecialTGenerationٮ89-ō>;%ExerciseO1.0 Giv9e)anexampleofatermordering)R,amoAduleMPH-=r, %andoasetofelemen9tsofg1*;::: ;gsgPH-=r7nMgwhichosatis esConditionsoA1*)%andTA2*).Wo%Exercise8 2.LetdKbAeda eld,letP^=K[x1*;::: ;xn7],letcbAeaterm%orderingioniT-=n7,letg1*;g2 R:2'PLbAet9woK-linearlyiindepAendentlinear%pAolynomials,xandletxi1*;i2p2Ff1;::: ;ngbAexsuc9hthatxi'Ӱ1 Y=FL:Tm(g1*)and%xi'Ӱ2 X=L:TBvF1(g2*).TPro9vethatthefollowingconditionsareequiv|ralent. *?a)7xConditionsTA1*)andA2)holdTforg1*,g2.)b)8xxi'Ӱ1 X6=xi'Ӱ2%Exercise3.DPro9ve/thatfor/rOP= 1and`3=RevLex!g ,/ConditionsB1*)%andTB2*)areTstrictlyw9eakerTthanTA1)andA2).%Exercise4.KLetfr=91and"=DegLex!).fSho9wthatthepAolynomials%g1h=>x1*x2}Sx2Rand|g2=x-=2h1}Sx2Rdo|notha9ve|propAerties|B1*)andB2).%Findr&L:T"֟DegLex$((g1*;g2))r&andr&athirdpAolynomialg3 2#(g1*;g2)suc9hr&that%fg1*;g2;g3gTsatis esB1*)andB2).%ExerciseGD5.Let>bAe>atermorderingonT-=2*,andletg1\=x-=3>ԗ1%andƒg2=-yR-=3ayR.ƒPro9vethatƒfg1*;g2gsatis esB1)andB2).ƒRepresen9t%fq=x-=3*y+xyR-=33Yx-=3xyy+1=0;:::;d,letai62KObGethecoecientofԵx^i intheminimal%pGolynomialB!ofB!`overB!K=andhi 2QK[x]theremainderofthedivision%ofQg[ٟ^ivbyf.QProvea0M+ڵa1|sh1+[+adhdpʲ=0QandQshowthatthisyieldsa%systemofdlinearequationsfora0|s;:::;ad.Explainhowwecanuseits%solutionUUspacetoanswerourquestion.\Ic)%Implement8themethoGddevelopGedinb)inaC0oLCoAfunction8LinAlgMP-8˲(::: )%whichTtakesTfandg -andcomputestheminimalpGolynomialof`overTK.%Hint:Y*oumayusetheC0oLCoA nfunctionSyzd(::: )to ndthesolutionspace%ofUUasystemoflinearequations.ZʞWō>;904`2.Gr`obnerTBases-ō>;?ֲd)%ApplyyourfunctionLinAlgMP-(::: )tocomputetheminimalpGolynomials %overUUQofUUthefollowingalgebraicnumbGers.)1) 919&fes2AA+ 'il&fes2 FaPpPfeE3 E)2);?Zt49PpAmPfeE2H+8Pp 7PfeE2+82)3)8x(x^3 +oQxoQ1)=x,pwherepfڧ=x^5ĸx2.(Hint:Noticethat 1&fe}Ux &= K1K&fes2 ̜ )x^4Ѹ 1&fes2*.)\Ie)%NoweweconsidertheidealeȵI=(x2=Yʵg[ٲ(x1|s);f(x1))K[x1;x2].eProvethat%a pGolynomial h2K[x2|s]satis esh(`)=0if andonlyifh2I. Conclude%thattheminimalpGolynomialof`overKDisanelementofminimaldegree%inUUtheprincipalidealUUI¸\8K[x2|s].(Hint:ShowthatK[x1|s;x2]=IT͍+3=0L.)f)%ProvemthatmL*T%Lex.(I)mcontainsapGowerofmx2|s.Concludethat,inorderto% ndtheminimalpGolynomialof`overK,itsucestocomputeasystem%ofzgeneratorsofzICwhichsatis esConditionsB1|s)andB2)withzrespGect%toUULex.g)%W*riteaC0oLCoA"9functionLexMP(::: )whichtakesfandg޲andcomputes%thepminimalpGolynomialofp`overpK:usingpthemethoddevelopedinf).%(Hint:\fY*oumayassumethatthebaseringis\fQ[x[1];x[2]];LexLFandapply%theC0oLCoA"/functionLT(I) &.)UseyourfunctionLexMP(::: )tocheckyour%resultsUUind).?h)%Compute4theminimalpGolynomialof4(x^3 *+Xx ̸X1)=x^5h overQin4thecase%f=zx^7zx1usingbGothLinAlgMP.(::: )andLexMP(:::).W*ritedown%thetwopGolynomialswhoseleadingtermsgenerateL*TQsLex|(I)Z=(x1|s;x^7l2).%WhichUUofthetwomethoGdsisingeneralmoreecient?Why?i)%Developldi erentmethoGdsforcomputingtherepresentationofl`^1(in%theUUK-basisf1;MSxa;:::;MSx^d1GgofLusingUUthefollowingideas.)1)7xLinearUUAlgebra)2)7xTheUUExtendedEuclideanAlgorithm%Prove[thecorrectnessofyourmethoGds.ThenwritetwoC0oLCoA!ܗfunctions%LinAlgInvT(::: )L andExtEucInv2(:::),L andcomparetheresultsinthecases%ofUUd).[Wō>;b2.2RewriteTRulesٮ91-ō>;2.2**RewriteRulesLAÎlxlNx^2|sy[ٲ,mg1 U=x^2b x+1,g2 U=>xyhȸxy+3,mandletm5=>DegLex%Y,.Sincex^2|syisamultipleofL*T(:в(g1|s),the rststepoftheDivisionAlgorithmappliedtofoand(g1|s;g2)yieldsf<9=(y_g1Z+0g2+(fsvy[g1|s),andwe nd궵fy[g1=uy(x1).ٍInthis rststepwehavereplacedٍfbyfy[g1|s.ٍThecorefofthisopGerationistotakefg1|s,writeitasx^2D}(x1),fandreplacex^2궲byYcx;1.YcThusweuseYcg1ֲasaruleforreplacingitshe}'ad,namelyx^2|s,byitstail,namelyxm1.Clearly*,ifapGolynomialg1 iswrittenasamb,wehave궵a=bAmoGd(g1|s),AbuthereweemphasizethefactthatAa=bAmoGd(g1|s)AcanAbGeviewedDasaruleforreplacingDabyb.DInotherwords,weorienttheequalitybyUUdestroyingitssymmetryinordertouseapGolynomialasar}'ewriterule."NowwecontinuewiththeDivisionAlgorithm.FirstweobservethatL*T E%۲(xyy[ٲ)x=xyjisGamultipleofGL*T M(g2|s).Sothesecondstepyields궵fg=Sxy3Fmg1S+1g2+(fy[g1g2|s)C)andfmy[g1g2=Sxx3.C)AgainwestressfthepGointthatthecoreofthisoperationistousefg2&asarewriteruleinthesensethatitsleadingtermxyisreplacedbyitstailxb+y3.HeretheUUDivisionAlgorithmstops."SuppGoseinsteadthatweperformtheDivisionAlgorithmwithrespectto ݵf land(g2|s;g1). Thenweget ݵfڧ=(x+1)g2$d+1g1+(f(x+1)g2g1|s),andweseethatݵfot[(x+1)g2Xg1=x+y4.ThealgorithmstopsandreturnsUUanoutputwhichisnotthesameasbGefore."Summarizing,:wecansaythatthecoreoftheDivisionAlgorithmistousetheIelementsIg1|s;:::;gsZasrewriterules.T*ousegihasarewriterulemeanstoreplace{Htheleadingtermof{Hgiϔbytheremainingpartofit,withtheobviousadjustmentX;924`2.Gr`obnerTBases-ō>;궲may6leadtodi erentresults.Sothenaturalquestioniswhetherthereare setsdAofrewriterulessuchthatallpGossiblepathscanbecontinueddAuntiltheyreach׬thesameresult.\Con uence"isthenameofthisgameandtheessenceofUUthissection,amoGdernversionofthemotto\allr}'oadsleadtoRome"."And9thereisa nalsurprisingresult.W*ewilldiscover9thatforasetofpGolynomials-orvectorsofpolynomials,beingspecialinthesenseofcon uenceis'equivqalenttobGeingspecialinthesenseofConditionsA)'andB)describGedinSection2.1.Thusrewriterulesprovideadi erentaspGectofthesamephenomenon.,uAlthoughitisbGeyondthescopeofthisbook,itturnsoutthatthis3viewismostsuitableforgeneralizationsinanumbGer3ofdirections,e.g.toUUthenon-commutativeUUcase."Nowitistimetostudytheseideasinamoretechnicalmanner.LetᎵKbGea{ eld,{n6=1,P*=K[x1|s;:::;xnq~]a{pGolynomialring,r51,andnTamoGduletermUUorderingonUUT^nq~he1|s;:::;ermi.!De nitionT2.2.1.is~LetUUg1|s;:::;gs R2Pc^r n8f0gandG=fg1|s;:::;gsF:g.a)%Letcm1|s;m2 /2Pc^r&,candsuppGosethereexistaconstantc2K,caterm%tȸ2T^nq~,pandanindexpiȸ2f1;:::;sgsuchpthatm2 ;=m1rfctgi?and%tdbL*Tj(giTL)PO=3ݸ2 fSupp#KN(m2|s).Thenwesaythatm1  reduces ato am2 inone%stepEPusingtherewriterulede nedbyEPgi(orsimplythatm1òreduces %tom2 sinonestepusinggiTL),andwewritem1ꍑ }gi$' C!:m2|s.Thepassage%fromUUm1Ȳtom2isUUalsocalledareductionTstep. k?b)%Thetransitiveclosureoftherelationsꍑg1$θ D! E;:::;ꍑآgs$!iscalledtherewrite󻍑%relationYXde nedbyYXGandYXisdenotedbyT΍ JG2YX u!uϲ.Inotherwords,fore%m1|s;m2 θ20[Pc^r&,weletm1T΍ jG2' C!:m2 wZifandonlyifthereexistindices%i1|s;:::;itLn2f1;:::;sgUUandUUelementsm^0l0|s;:::;m^0፴tLn2Pc^r &suchUUthat3r<ڵm1C=m0፱0[gi1G' C!:m0፱1[gi2G' C!( -giZjt"%C+_!7 m0፴tLn=m2\Ic)%An?element?m12Pc^r pwiththepropGertythatthereisno?i2f1;:::;sg%andnom2C2Pc^rVngfm1|sgsuchthatm1ꍑ }gi$' C!:m2x iscalledirreduciblewith󻍑%respGectUUtoT΍ FG2UU q!q̲.e?d)%TheUUequivqalencerelationde nedbyT΍ FG2UU q!!willbGedenotedbyT΍ cG2UU UX!."InIparta)ofthisde nition,wecanchoGoseIc^=0andt2T^n !DzsuchIthat궵t L*T W1(giTL)㊵=2 8ܲSupp"q(m1|s).Thisiscalledatrivialreduction.Byusingitweseethatlm1ꍑ }gi$' C!:m1|s.lIntheexamplementionedintheintroGduction,wehaveforeinstancetfꍑ兴g1$7ڭ!Vxy5!y/Mandxy5!yꍑ-ϴg2$"!͠x3.tThustfT΍G27ڭ!Vx3andx3T΍hG2 UX!f궲hold,whileĵxո3T΍G2!qǵfSisnottrue,bGecausetheleadingtermofĵfislarger thanUUx.][Wō>;b2.2RewriteTRulesٮ93-ō>;PropQositionT2.2.2.q(PropQertiesTofRewriteRelations) L}'etg1|s;:::;gs R2Pc^r n8f0g,andletG=fg1|s;:::;gsF:g.󻍍~oa)%Ifm1|s;m2C2Pc^rpsatisfym1T΍ jG2' C!:m2 Zandm2T΍ jG2' C!m1|s,thenm1C=m2.󻍍Hb)%IfFm1|s;m2 k2'Pc^r satisfym1T΍ jG2' C!:m2|s,FandifFt2T^nq~,thenwehave%tm1T΍ jG2' C!:tm2|s. 󺍍Hc)%Everychainm1T΍ jG2' C!:m2T΍ jG2' C!'suchthatm1|s;m2;:::g2Pc^r b}'ecomeseven-%tuallystationary. k~od)%If)m1|s;m22Pc^r Zsatisfym1ꍑ }gi$' C!:m2 fori2f1;:::;sg,)andif)m32Pc^r&,%thenFther}'eexistsanelementFm42 mPc^r !suchthatm17+m3T΍ jG2' C!:m4 cand 󺍑%m2S+8m3T΍ jG2' C!:m4|s.He)%Ifm1|s;m2;m3;m4 U2Pc^r satisfym1T΍ 4۴G2' UX!'m2 U^andm3T΍ 4۴G2' UX!m4|s,thenwe󻍑%havem1S+8m3T΍ 4۴G2' UX!'m2+m4|s.~f )%Ifm1|s;m2 32Pc^r satisfym1T΍ 4۴G2' UX!'m2|s,andiff2Pc,thenwehave%fm1T΍ 4۴G2' UX!'fm2|s.eHg)%F;orm2Pc^r&,wehavemT΍hG2 UX!0ifandonlyifm2hg1|s;:::;gsF:i.~oh)%F;orm1|s;m2 2Pc^r&,wehavem1T΍ 4۴G2' UX!'m2 ifandonlyifm12{m2 2 %hg1|s;:::;gsF:i.Pr}'oof.6T*opshowclaima),weconsiderachainofreductionstepswhichrepre-sents]m1T΍ jG2' C!:m2T΍ jG2' C!m1|s,]i.e.achain]m1C=m^0l0[gi1G' C!:( -giZjt"%C+_!7 m^0፴tLn=m1вsuch]that궵i1|s;:::;itՄ2P.f1;:::;sgandm^0;Zjڲ=P.m2 $ forsomej⹸2f1;:::;to1g.Thee ectof0areductionstepisthatatermisreplacedbyotherterms,allofwhichare]tsmallerwithrespGectto]t[ٲ.Solettek Iwithtԡ2T^n andk%82f1;:::;sgbGetheEilargesttermwithrespGecttoEiBwhichisreducedinthischain.ThistermisgFnotcontainedinthesuppGortoftheresultanymore,unlesseachreductionstepUUistrivial,i.e.unlessUUm1C=m2|s."ClaimFb)holds,sinceitholdsateachreductionstep.Thusweprovec)now.SuppGosethereexisti1|s;i2;:::AW2f1;:::;sgandm1|s;m2;:::AW2Pc^r Jsuch3that7wehaveachainofreductionsteps7m1[gi1G' C!:m2[gi2G' C!(Ќwhich7doGesnotbGecome)stationary*.The rstclaimisthateach)mi +umust)haveaterminits%suppGortwhichreduceseventually*.IndeedweobservethatifthisdoGesnothappGen,itmeansthatstartingfrommi thesequenceofreductionsisactuallyaYsequenceofequalities.ThereforethereexistsatermYtiFinSupp(miTL)whichYistheClargesttermwithrespGecttoCzwhichisreducedlaterinthechain.Thenwehavest1C)t2U,sandsinceeverytermstiӲisreducedeventually*,thischaindoGesUUnotbecomestationaryeither,incontradictionwithTheorem1.4.19."F*or,theproGofofd),welet,c'2K,t2T^nq~,,andi2f1;:::;sgbGe,suchthatm2@=͵m18hctginBandtL*TN7Ͳ(giTL)+?=͸2FSupp%.(m2|s).Clearlywemayassume궵cb6=0.W*elet쵵c^0bGethecoecientof쵵tL*TN7Ͳ(giTL)inm3 i(anddistinguishtwocases.3When3c^0=޸c,wehave3m1#<+ɵm3VQ=޵m2+m3+ctgi.*=޵m2+m3궵c^09tgiTL.jSincethecoGecientofjtL*TN7Ͳ(giTL)inm2춲+pCm3c^09tgivqanishes,jweget^eWō>;944`2.Gr`obnerTBases-ōeߍ궵m2+m3ꍑ }gi$' C!:m1+m3|s,%andwecanchoGose%m4=m1+m3|s.Whenc^0`U6=c 궲weUUde neUUm4Ȳby}TjRm4C=m1S+8m3(c+c09)tgid=m2+m3c09tgi궲andUUobtaintheclaim,bGecausethecoecientofUUtL*TN7Ͳ(giTL)vqanishesUUinm4|s."Next, Pclaime)followsfromd),andf)followsfromb)ande)byrepre-sentingfasasumofmonomials.Sinceh)isanimmediateconsequenceofe)󻍑andg),itremainstoshowg).IfmT΍hG2 UX!0,wecollectthetermsusedinthevqariousKYreductionstepsandgetarepresentationKYm=f1|sg1\+$y+$fsF:gswith궵f1|s;:::;fsp2*Pc.Conversely*,givenanelementm*2Pc^r bJwithsucharepresen-etation,itsucesbye)toproveֵfiTLgiT΍ G2 UX!0fori=1;:::;s.Thisfollowsfrom궵giT΍ G2 UX!0UUandUUf).k卑"궲Unfortunately*,itisnotclearhowwecouldusepartg)oftheabGovepropGosition*tocheck*whetheragivenelement*m*]2Pc^r is*containedinthesubmoGdulehg1|s;:::;gsF:i,becausewedonotknowthedirectionofthereduc-tionYstepsusedinYmT΍hG2 UX!0.Inotherwords,ifweuseonlyreductionsteps<궵m=m0[gi1G' C!:m1[gi2G' C!$,wemightgetstuckatsomepGointwithanirreducibleeelementwithrespGecttoT΍ G2 !.ThenextexampleshowsthatthiscanreallyhappGen.}ExampleT2.2.3.c;b2.2RewriteTRulesٮ95-ō/@C4|s))Ifµm1|s;m2;m3C2Pc^r satisfyµm1T΍ jG2' C!:m2U5andm1T΍ jG2' C!m3|s,thenther}'eexists󻍑)AanBelementBm4C2Pc^r bssuchthatm2T΍ jG2' C!:m4 andm3T΍ jG2' C!m4|s.B(AAr}'elationT΍.2G2)A/^!=withthispr}'opertyisc}'alledcon uen9t.)X۩r&Gm1hIJm3hIm2&Gm4PU;O line10PU\PUɱPU=IaIa3PUQPU{QPU&EQPUQIvzQIvzs˙vMB>ـ;8^Kn]n]3˙aVQŠKQB˙[Qـ;PQ8yQnQnsـ^GـWGaWGaǩGPr}'oof.6F*orGtheproGofofGC1|s))C2),GwenotethatifGm2M,GthenC1|s)implies}mT΍G2!qDz0.}ThusifmisirreduciblewithrespGecttoT΍̴G2 !,wegetm=0. NextweshowthatϵC2|s)impliesC3).ByPropGosition2.2.2.c,thereisanelement>m2C2Pc^r wowhich>isirreduciblewithrespGecttoT΍G2 ´!handwhichsatis ese궵m1T΍ jG2' C!:m2|s.ySuppGoseym^0l2C2Pc^r Jܲisanotherelementwiththoseproperties.Then&we;have;m2AFεm^0l2 ]2iM,sincem1T΍ jG2' C!:m2 fandm1T΍ jG2' C!m^0l2|s.;F*urthermore,the(element(m2Aϸ\m^0l2 isirreduciblewithrespGecttoT΍ ^G2 D!D,sincenoterminSupp*#(m2|s)[Suppq(m^0l2)isamultipleofoneofthetermsL*T97Ͳ(g1|s);:::;L*TN7Ͳ(gsF:).ByUUC2|s),UUweconcludem2C=m^0l2|s."NowNweproveNC3|s) )C4).NByPropGosition2.2.2.c,thereareelements󻍑궵m^0l2|s;m^0l3b2;964`2.Gr`obnerTBases-ōHb)%Byc}'ollectingallreductionstepsinmT΍G2!qDz0,wegetf^0l1|s;:::;f^0፴sh32!P)such %thatmVk&hy4LC S᪱(m)ʉfe?LC hv(g  )%tg Բ=Pލ -s% -i=1$f^0;ZiȵgiandsuchthatL*T`;Ѳ(m)>sL*Ty(f^0;ZiȵgiTL) [%fori=1;:::;swithf^0;Ziȵgid6=0. 񍍍Hc)%Ifweputfi߲=f^0;Zi ofori2f1;:::;sgnf zgandf du=f^0፴ +&hڱLCP(m)0Dʉfe?LC hv(g  )%^t,={%then zweobtainanelement zm=Pލ, s%, i=1KfiTLgi^whose zle}'adingtermsatis es %L*T2U.7IJ(m)=maxc$fL*T %(fiTLgi)ji2f1;:::;sg;qfiTLgid6=0g.Pr}'oof.6Claimka)followsimmediatelyfromthefactthatkL*TV(m)khastobGeeliminatedUUatoneofthereductionsteps."Nowtweproveb).Lettm1|s;:::;mtġ2?KPc^r nbGesuchthattm1=?Km,mtġ=0,󻍑and!forall!ik=1;:::;th1we!havemiT΍ CG2 j!mi+1Npusing!onereductionstep.By5a),thereexistsareductionstepwheretheleadingtermof5mis5reduced.Thisstepisunique,sinceitsubstitutesL*T2((m)withsmallerterms.So,let궵`2f1;:::;t81gUUbGeUUsuchthatm`+1=m`Ƹ&h[LCȟ(m)lʉfe?LC hv(g  )$-tg .Then"?궵m4<$LCg~(m)gwfe% (֍LC x{(g ),notg y=m(m`qm`+1 )=5P`1 X tմi=18(mi쀸mi+1 tO)+/t1 X i=`+1(mimi+1 tO) %is xoftheform xPލs%i=1f^0;ZiȵgiTL.HerethepGolynomials xf^0;Zi@areobtainedbycollectingthe@elementsoftypGe@ctappearing@inthetwo@sums,whereeachdi erence궵mijȵmi+1is1oftheform1mijȵmi+1*=C۵ctg for1some1c2K,t2T^nq~,1and궵 2f1;:::;sg.eT*oconcludetheproGofitsucestoobservethatwhenwewrite*mi7Pmi+1;g=ctg d,*wegettUPL*Tߟ[u(g d))L*T=/Ӳ(miTL)bythede nitionofaUUreductionstep."Finally*,UUweseethatc)isanimmediateconsequenceofb).8 "궲LetUUusexaminetheclaimsofthislemmainaconcretecase.ExampleT2.2.7.c;b2.2RewriteTRulesٮ97-ō>;"궲Unfortunately*,thelemmarequiresthattheelementreducestozero.In ourcase,wecouldhavefollowedadi erentsequenceofreductionsteps,forinstanceWV&x3ꍑ Nig1$' C!:x2|syꍑ-ϴg1$"!͠xy[ٟ2ꍑ Bg2$ j!Jxy+8y[zꍑBg2$?7!^yzw+x+z궲HerePweendupwithanelementwhichcannotbGereducedfurtherandwhichisnon-zero.ByloGokingatthissequenceofinstructionsteps,wecannotdecidewhetherUUx^3Ȳsatis esUUthehypGothesisofthelemma.|V"BothFintheintroGductiontothissectionandinthepreviousexamplewehave+seenthatthepropGertyofbeingcon uentisnotsharedbyallrewriterelations.IsthereabGetterwayofunderstandingit?ThefollowingpropositiongivesUUasomehowunexpGectedanswer.PropQositionT2.2.8.qL}'etg1|s;:::;gs 2ϵPc^rZn)f0g,letG=fg1|s;:::;gsF:g,andlet} M=mshg1|s;:::;gsF:i.} ThenConditionsA1|s),A2)of} Pr}'oposition2.1.1aree}'quivalentwithConditionsC1|s),C2),C3),andC4)ofPr}'oposition2.2.5.Pr}'oof.6T*oproveA2|s)')C2)bycontradiction,wesuppGosethatthereisan󻍑elementk~m2M knPf0gwhichk~isirreduciblewithrespGecttoT΍ \ʹG2 !.ByCon-ditionsA2|s),stheelementmhasarepresentationmI=Pލs%i=1{fiTLgifsuchsthat궵f1|s;:::;fs2̵PLQandL*TQ(m)=maxYBظfL*T %(fiTLgi)̸ji2f1;:::;sg;gtfiTLgi6=0g.LettL*TN7Ͳ(giTL)bGethetermwhichachievesthismaximum.Thentheelement궵m^0Q=m8&hlLC2ԉ(m)lʉfetLC k(gi*)!`Ytgisatis esUUmT΍G2!qǵm^0#andm^06=m,UUacontradiction. ˍ"Conversely*,UUC1|s))A2)UUfollowsUUdirectlyfromLemma2.2.6.4U90V%Exercise1.Leto(bAeoamoduletermordering,letogp2PH-=rnnf0g,andlet )%G=fgRg.TSho9wthattherewriterelationp G&T n!iscon uent.%Exercise72.Letր)*bAeրamoduletermordering,andletրGbeրa niteset %of\termsin\PH-=ruS.Sho9wthatConditionsC)ofPropAosition2.2.5holdforthe%rewriteTrelationp G&T n!j.)%Exercise V3.qGiv9eanexampleofarewriterelationp cG& !rwhichisnot%con uen9t.%Exercise24.Let.a bAe.aamonoidorderingonT-=n7,lett1*;t22f1T-=n LbAe%termspwithpt1>t2*,andletg=Jt1Gt2*.Considertherewriterelation%de ned>b9y>Gۡ=fgRg.(Observ9ethatherewedonotassumethat>-isa%termTordering.)Pro9veTthatthefollo9wingconditionsareequiv|ralent. *?a)8xt1m-t2 \)b)7xEv9ery&ychain&yf1p MG&Q \k!'f2p MG&Q \k!%0suc9hthat&yf1*;f2;:::2YPo\bAecomes&yev9en-7xtuallyTstationary:.bo(Wō>;984`2.Gr`obnerTBases-ō>;T utorialT18:AlgebraicNum9bQers궲Inuthistutorial,wewanttouseC0oLCoA$togivesomehintsabGouthowone can e ectivelycomputeinthe eld CfeQ^ofalgebraicnumbGers,i.e.thealgebraicclosure of Q.W*eshallcomputeonlyuptoconjugates,i.e.weshallrepresentanؾalgebraicnumbGerؾbyitsminimalpGolynomialoverؾQ.T*odistinguishbGe-tween`conjugatealgebraicnumbGers,`wewouldalsohavetoprovidereasonablygoGod>approximationsin>Q[i].F*urthermore,weshallbGecontentto ndsome궲pGolynomialwhichhasacertainalgebraicnumbGerasoneofitszeros.AfterfactoringOthispGolynomialusingtheC0oLCoA#!functionOFactor(::: )oneOcouldthen7trytousemethoGdsofnumericalanalysisto ndthefactorwhichistheminimalUUpGolynomialofthedesiredalgebraicnumber."Letqa1|s;a2C2CfeQbGeqtwoalgebraicnumbGersrepresentedbyirreduciblepGoly-nomialsUUg1|s;g2C2Q[x]ofUUdegreesd1;d2|s,UUrespGectively*.a)%Use#Macaulay'sBasisTheorem1.5.7toshowthattheresidueclassesof%fx^il1|sx1ɍjxݍ2 ajK0i; O2.3Syzygiesٮ99-ō>;2.3**SyzygiesNotN;1004`2.Gr`obnerTBases-ō>;궲terms)-canbGeliftedifandonlyifthesetofgeneratorssatis esConditions 궵A),UUBq),UUandC)!{De nitionT2.3.1.is~LetsSRbGesSaring,MnanRDz-moGdule,andG=(g1|s;:::;gsF:)aUUtupleofelementsofUUM.a)%ARsyzygyRaofRaG2isatuple(f1|s;:::;fsF:)2Rǟ^sbsuchRathatf1|sg1o+2 +2fsF:gs R=0.?b)%ThebJsetofallsyzygiesofbJGformsanRDz-moGdulewhichwecallthe%( rst)syzygymoQdule²of¸GandwhichwedenotebySyzDqƴRC(GѲ)orby%Syz4qƴR; (g1|s;:::;gsF:).KIfnoconfusioncanarise,weshallalsowriteKSyzej(GѲ)Kor%Syz4(g1|s;:::;gsF:).3S"AsVintheprevioussections,weletVKRrbGea eld,n1,P*=K[x1|s;:::;xnq~]anpGolynomialring,nr8y\1,andʊamoGduletermorderingonT^nq~he1|s;:::;ermi.F*urthermore,weletg1|s;:::;gs R2Pc^rln;f0g,weletM3=hg1|s;:::;gsF:iPc^r{,andwe#denotethe#s-tuple(g1|s;:::;gsF:)byGѲ.#Thenweconsiderthe#Pc-moGdule궵Pc^s ,withcanonicalbasisf"1|s;:::;"sF:gandthehomomorphismɲ:Pc^s!M궲givenhGbyhG"jW7!櫵gjforjy6=1;:::;s.hGInthissituationwecanalsodescribGethesyzygyUUmoGduleofUUG&bySyztqƴPGc(GѲ)=ker#()."Thenatureofmanyfactsexplainedinthissectionisnotelementary*,sotheinexpGeriencedreadermighthavesomediculties.F*orinstance,itisclearthatWevenifwestartwithanideal,givenbyasetofpGolynomialgenerators,thesetoftheirsyzygiesisamoGdule.SothetheoryisdescribedintheframeworkofmmoGdules.Moreover,mweshallneedtointroGducea negradingonthemoduleof:syzygiesinordertodetectthecorrect\highesthomogeneouscompGonent"whenUUwefollowtheabGoveapproach."SinceTdwedonotwantanyreaderrunningawayfromthisbGookTdatthispGoint,wedecidedtouseadidactictoGol:arunningexample.Thisisanex-amplewhichwewillrevisitseveraltimesduringthesection,andwhichwewill:"usetomakeallde nitionsandconstructionsaslucidaspGossible.LetusstartUUourrunningexamplebyintroGducingitsbasicob8jects.ExampleT2.3.2.c!ȵM Y!Pc^ro!Pc^r1=qM!0with44thedescriptionof44Syz{S(GѲ)44asthekernelof,weobtainalongexactsequenceI00 !0SyzO(GѲ)0!0PcsT΍ʨ2 7!Pcr _a{!Pcr1=qMK !0eWō>; O2.3Syzygies9101-ō>;"궲NowletN -Pc^r bGethePc-submoGduleofPc^r generatedbythevectors 궸fLMjC(g1|s);:::;LMUw(gsF:)g,kletkLM.(GѲ)kbGethetuple(LMjC(g1|s);:::;LMUw(gsF:)),and{let{:Pc^spW!Ndenotethehomomorphismgivenby{"jĸ7!LM1ş[(gj6)for궵jt=51;:::;s.ThenKerb()isthesyzygymoGduleofLMe(GѲ).Consequently*,itUUwillbGedenotedbyUUSyzt(LMjC(GѲ)).W*eobtainanotherlongexactsequence󻍑;1024`2.Gr`obnerTBases-ō>;궲homomorphismPc^re!Pc^r1=qNԲisahomomorphismofT^nq~he1|s;:::;ermi-graded 궵Pc-moGdules)ZbyRemark1.7.9.Thusthewholesequenceconsistsofhomomor-phismsUUofUUT^nq~he1|s;:::;ermi-gradedmoGdules.$Example2.3.2(con9tinued))tInourexamplewehave)tLM!(GѲ)=(x^2|s;xy[ٟ^2L).Then=forinstance=(Pc^2)x2 y@L2ò=f(c1|st1;c2t2)2Pc^2jc1t1x^2;c2t2xy[ٟ^2d2Qlx^2|sy[ٟ^2Lg.Examples+7ofelementswhichbGelongto+7(Pc^2)x2 y@L2Pare(y[ٟ^2L;0),(y[ٟ^2;x),+7and( 33133&fes2bٵy[ٟ^2L;4x). J"The^intrinsicmeaningofthenewconceptswhichwearenowgoingtointroGducemnwillbediscussedmorethoroughlyinV*olume2.ForthetimebGeing,*theyareonlyde nedwiththepurposeofbetterdealingwiththeT^nq~he1|s;:::;ermi-gradingsUUdescribGedabove.!:De nitionT2.3.4.is~Letu2mbGeu2anon-zeroelementofau2T^nq~he1|s;:::;ermi-gradedmoGdule,Q*andletQ*m=P US2Tnlhe1 ;::: ;eriE0m beQ*thedecompositionofQ*mintoQ*its @homogeneous@compGonents.Theterm@max5Lf8J2T^nq~he1|s;:::;ermijm 6=0g@iscalled$the$[-degreeofm,andthehomogeneouscompGonentofmofthisdegreeUUiscalledtheUU[-leadingTformofm."InthecaseoftheT^nq~he1|s;:::;ermi-gradingonPc^s @de nedinPropGosi-tion42.3.3,wedenotethe4[ٲ-degreeofanelement4m2Pc^sinf0gbydegßqƴ;Gj3(m),andits[ٲ-leadingformbyLFx;GP(m).InthenextpropGositionweshowhowtoUUdetermineUUdegxqƴ;G T(m)UUandLFt;G(m)foranon-zeroelementm2Pc^sɲ.PropQositionT2.3.5.qL}'et;themodule;Pc^s tbeequippedwiththe;T^nq~he1|s;:::;ermi-gr}'ading=)de nedabove,let=)f1|s;:::;fs R2Pc,andletm=Pލ USs% USjg=1Vfj6"jĸ2Pc^s&n|Of0g.~oa)%Wehavedegvqƴ;G B(m)=maxc$fL*T %(fj6gj)jjY2f1;:::;sg;qfj6gjĸ6=0g. HHb)%WehaveLF[;Gv(m)=Pލ USs% USjg=1\q}˲Vfj&r"j6,wher}'e$+捍\q2?0fj;=8 >< >:ȍ UO0'if'εfjIJ=0orL*T7v (fj6gj)<)degM=qƴ;G#ح(m) UOcj6tj'ifL*T7v (fj6gj)=deg꧟qƴ;Gv(m)andcjĸ2K,tj2Supp(fj6)'ar}'esuchthatLMa*(fj6gj)=cj6tjTLML(gj)"OrPr}'oof.6ClaimXa)followsfromPropGosition1.5.3andDe nition2.3.4.T*oshowGb),weusethatGdegjqƴ;G(m)=maxc$ftL*TN7Ͳ(gj6)j1jYs;qt2Supp(fj)g궲byPa),andthismaximumisachievedpreciselyforthetermsdescribGedintheformula."궲Sometimes~3wearedealingwiththecase~3rRR= 51,orwecanpickamonoidorderingonT^n Ihsuchthat3òiscompatiblewith!Dz.Inthiscase,wehave\q궵fjѲ=cj6tjIJ=LM1ş0(fj)UUinUUpartb)ofthispropGosition. JExampleԊ2.3.2(con9tinued)3ALetuscomputebGoththe3A[ٲ-degreeandthe궵[ٲ-leadingAformbofsomeelementsofbPc^s Z+inourrunningexample.F*orinstance,if_weconsiderthepair_( 33133&fes2bٵy[ٟ^2Lzp;4xz),_wehave_degqƴ;G z^( 33133&fes2bٵy[ٟ^2Lzp;4xz)=x^2|sy[ٟ^2z A궲andWLFv;G1( 33133&fes2bٵy[ٟ^2Lzp;4xz)=( 33133&fes2y[ٟ^2Lzp;4xz).WAlternatively*,ifwestartwiththepair(y[ٟ^2LzkIx;4x^2wyW"3)2Pc^25Z,wegetdegqƴ;Ge(y[ٟ^2LzkIx;4x^2wyW"3)=x^3|sy[ٟ^2궲andUULFt;G(y[ٟ^2Lzw8x;4x^2Sy3)=(0;4x^2|s).gվWō>; O2.3Syzygies9103-ō>;"궲OurCnextgoalistoconnectthetwoClongexactsequencesconstructed abGove.*W*ede neamap*LM:*Pc^r _!Pc^r1,*whichsends*0to0andmtoLM#Uc((m)Lifmc6=0.LAnalogouslywede neamapLLFw:cPc^s [)!Pc^s whichsendsi0to0andmtoLF݈;Gh(m)ifm6=0.iInthiswayiwegetthefollowingfundamen9talTdiagram.&|K$0396![Syzj(GѲ)cٸO!QPc^sT΍D2Dz!۲!Pc^r W! YPc^r1=qM17!K05?5?5yDSLF{+?{+?{+yD%ױLM$0396!M6Syz\}(LMjC(GѲ))cٸO!QPc^sT΍92Dz!۲!Pc^r W! ௵Pc^r1=qN17!K0"Thisdiagramsuggestsnaturalquestions,forinstancewhethertheverti-calmapsarehomomorphisms(clearlytheyaren't),andwhetherthediagramcommutesGG(itdoGesn't).AGCmorepreciseanswertothesecondquestionispro-videdUUbyournextpropGosition.PropQositionT2.3.6.qInthesituationdescrib}'edabove,letm2Pc^s⩸n8Syz(GѲ).~oa)%WehaveL*T7v ((m)))degM=qƴ;G#ح(m).Hb)%We8have8LFUW(m)2Syz7(LMjC(GѲ))8ifandonlyifL*T1ǟ]((m))<)degM=qƴ;G#ح(m).Hc)%Weihavei(LF (m))=LM1((m))ifiandonlyifL*T[((m))=deg꧟qƴ;Gv(m).Now,letm2Syz7(GѲ)inste}'ad.~od)%WehaveLF[(m)2Syz7(LMjC(GѲ)).Ther}'eforethemapLFinducesamapsLF!}j:Syz ,v:(GeA)ꣲ:Syz7(GѲ)!Syz(LMjC(G))%whichwedenotebyLFagain.Pr}'oof.6Claimoa)followsfromtherulesforcomputingwithleadingterms (see_PropGosition1.5.3)andfromProposition2.3.5.a.Namely*,fortheelement궵m=Pލ USs% USjg=1Vfj6"jĸ2Pc^s⩸n8f0gUUweUUcalculateNL*T"ݟ's((m))I*=SL*T`je(;s :Ptjg=1Wfj6gj))max$ (fL*T %(fjgj)jjY2f1;:::;sg;qfjgjĸ6=0g卍I*=Sdegcjqƴ;Goڲ(m)"T*oproveb),wewrite︵ms=PލJs%Jjg=1 fj6"j 2Pc^s?nJvf0gandLFן;GBG(m)= 궟Pލxs%xjg=1\q0i.zHfj7K"j!aszuinPropGosition2.3.5.Thenzu(LF (m))=Pލ 2s% 2jg=1\qfj([4LM7=(w(gj6)=0 g$is9equivqalenttothevanishingofthecoGecientof9degȟqƴ;G i8(m)9inPލHts%Htjg=1I˵fj6gj6,i.e.UUitisequivqalenttoUUL*T[z((m))<)degM=qƴ;G#ح(m)."T*oprovec),wenotethatL*T0H޲((m))#h6=degFqƴ;Gg(m)impliesbya)andb)thatKwehaveK(LF (m))b=0.KSince(m)6=0,KwethengetLM((m))b=LM#Uc(((m))A6=0=(LF (m)).Conversely*,ifLTBY((m))A=degeqƴ;G(m),then HLM#Uc((m))=LM1ş[(Pލ ;s% ;jg=1fj6gj)=P USfjgj'㍴fj6=0g/LM>DXR(fj6gj)=Pލ USs% USjg=1\q}˲Vfj(ULM7<꘲(gj6)= x捑궵(LF (m)).hWō>;1044`2.Gr`obnerTBases-ō>;"궲Finally=weshowclaimd).Let=m=Pލ-Դs%-jg=1/+fj6"jE2Syz(GѲ)znf0g.=Starting g$with(m)#=0,wegetthatthecoGecientofdegsqƴ;G ;(m)inPލs%jg=1vfj6gjÐvqan- ishes,uandhenceuPvfjgj'㍴fj6=0g0lLM@Ey(fj6gj)=Pލ USs% USjg=1\q}˲Vfj(ULM7<꘲(gj6)=(LF (m))=0. X D"궲LetUUuscheckUUtheclaimsofthispropGositioninourrunningexample.;+ExampleD2.3.2(con9tinued)landthus>deg͟qƴ;G!r=(m)(=x^2|sy[ٟ^2 is>notascalarmulti-%pleRofRLMD((m))Of=LM((m))=xzp^3 .Goingtheotherwayinthe%fundamentalfsdiagram,wecalculatefsLF-(m)=(y[ٟ^2L;x)fsand(LF (m))=%y[ٟ^2LMP7(g1|s)CxLMUw(g2)=0.ڸInparticularڸLF(m)2Syz7(LMjC(GѲ)).ڸHere%wePhaveacasewherePL*Tߟu((m))<)degM=qƴ;G#ح(m)PandwhereLM]((m))6=%(LF (m)).?b)%Thehelementhȵm灲=(x;y[ٲ)ofPc^2 Hʲsatis es(m)=xg1J+E׵y[g2c=x^3E׵xy[ٟ^2#%x^2ڲ+0gxy[ٟ^3y[zp^3 ,ȡandthusȡdeg0qƴ;G!w(m)1=xy^3 asȡwellasȡLM3N((m))1=%xy[ٟ^3L.Ontheotherhand,wecalculateLFO(m)=(0;y[ٲ)and(LF (m))=%yLMq.IJ(g2|s)=xy[ٟ^3L.WHerewehaveacasewhereWL*T|((m))=degߩqƴ;Gk(m)%andUULM((m))=(LF (m))."InGthisexampletheelementGm=(y[ٟ^2L;x)satis esm=2 8ܲSyz(GѲ),GwhereasLF (m)2Syz7(LMjC(GѲ)).ThefactthatLF:(m)isasyzygyofLMDȟ^(GѲ)maybGeconsideredoasasortof rststepintheconstructionofasyzygyofoGѲ.ThusapGossibleapproachtoourproblemofcomputingasystemofgeneratorsfor!Syzi(GѲ)!could!bGeto ndelementswhichgenerate!Syzi(LMjC(GѲ))!andto\lift"+themtoelementsof+Syz-J(GѲ)+insomeway*.TheremainderofthissectionisTudevotedtostudyingthefeasibilityofsuchanapproach.Asa rststepweseeUUhowtoobtainanexplicit nitesetofgeneratorsofUUSyzt(LMjC(GѲ)).덍TheoremT2.3.7.dH(SyzygiesTofElemen9tsofMonomialMoQdules)F;orĵj Z=w1;:::;s,wewriteLMqP(gj6)intheformLMqP(gj6)w=cjtje j zwith궵cj 2[K,4"tj2T^nq~,4"and4" j2f1;:::;rGg.4"F;orall4"i;j{2[f1;:::;sg,wede ne.덑궵tij =텍Klcm](ti*;tja)Kŭfe! ⦴ti(#.~oa)%F;orkallki;jE2f1;:::;sgsuchkthati; O2.3Syzygies9105-ō𐶍e%=nA}lcm}ϸ(tiTL;tj6)e j v=텍lcm(ti*;tja)ŭfe! tj*xgL*T7<~(gj)q=degOqƴ;G (tjgi "j6)"Nowweproveb).Inviewofa),itisclearthatSyz(LMjC(GѲ))6=0if andponlyifthereexistpi;jz2rf1;:::;sgsuchpthatiris>ahomomorphismofT^nq~he1|s;:::;ermi-gradedPc-moGdules,itskernelisaT^nq~he1|s;:::;ermi-graded@submoGduleof@Pc^s 겲andhasahomogeneoussystemofgenerators.!Letusconsideroneofthosehomogeneousgeneratorsandwriteit>Uas>UmKk=Pލ٦s%٦jg=1aj'Z<b6tj ʵ"j2Pc^s}n4f0gwithaj2Kqand'abtj\޸2T^nq~.>UThereareanindex2f1;:::;sgandatermt2T^n suchthat'ˢbtj%زL*Tg!+(gj6)=te궲whenever&εajZ6=$60,&whichisanotherwayofsayingthat&εmishomogeneousandmdegWqƴ;G!Dz(m)=te\.mNext,letmsize :(m)mdenotethecardinalityoftheset궸fi2f1;:::;sgjaid6=0g.Since(m)=0,wehavePލs%jg=1aj6cjIJ=0,andsince궵m6=0,.&itfollowsthat.&sizeʘ(m)2..&Hencethereareatleasttwo.&indices.& z; 궲such7~that7~a 6=@0anda 6=0.7~F*romt='cbt Yt ='cbt A]t dwe7~seethat7~tis7~amultipleUUofUUlcm(t ;t d),hence'C6gbC׵t O)C= ѴtK&febt  = +tK&fe$lcm c(t  ;t ʱ)*t ɲand'2-$b2 t == U0tK&fe t = +tK&fe$lcm c(t  ;t ʱ)t \ u궲W*ededucethatthesyzygy t&fe$lcm c(t  ;t ʱ)*[ Ihasthesameߵ[ٲ-degreeasm.More-overkweseethatifkm^0Q=mYa c  t&fe$lcm c(t  ;t ʱ),^ Ȳ,thenksize(m^09);1064`2.Gr`obnerTBases-ō>;"궲The nextstepsinourprogramaretogiveameaningtotheproGcessof \lifting"jasyzygyofjLMz(GѲ)jtoasyzygyofGѲ,andthentostudywhethersuchUUliftingscanalwaysbGefound.De nitionT2.3.9.is~AnRelementRm2Pc^s _iscalledaliftingofanelement궟~feǷgm y2Pc^s ifUUwehaveUULFt(m)=~feǷgm ϲ.PropQositionT2.3.10.wThefollowingc}'onditionsareequivalent./@D1|s)*ֹEveryhomo}'geneouselementofSyz(LMjC(GѲ))hasaliftinginSyz(GѲ)./@D2|s)*ֹTher}'e_existsahomogeneoussystemofgeneratorsof_²Syz(LMjC(GѲ))_con-)Asistingentir}'elyofelementswhichhavealiftinginSyz(GѲ)./@D3|s)*ֹTher}'e+existsa nitehomogeneoussystemofgeneratorsof+SyzVJ(LMjC(GѲ)))Ac}'onsistingentirelyofelementswhichhavealiftinginSyz(GѲ).Pr}'oof.6Since%DD1|s)!)D3)%Das%DanimmediateconsequenceofTheorem2.3.7,and4hsince4hD3|s):)D2)4hholds4htrivially*,itsucestoprove4hthatD1|s)followsfrom~D2|s).~LetIG}bGeaset,letf~feǷgmǷi gi2IM bGeahomogeneoussystemofgeneratorsofzSyz (LMjC(GѲ))zindexedzoverI,zandletmi2eSyz(GѲ)bGealiftingof~feǷgmBi궲foreveryi!ϸ2I.Givenahomogeneouselement~feǷgm~2!ϲSyzh(L*T %(GѲ)))nf0g, ctherej+existsanaturalnumbGerj+hsuchj+thatwehavej+~feǷgm=ҟPލ x h% x jg=1ydcj6tj~feǷgm cijy;with궵cj뼸2KO,nf0g,withtj2T^nq~,andwithij뼸2IforjG=1;:::;h.Clearly*,wemay*Passume*PdegMߟqƴ;GO(tj6~feǷgm cij)=deg꧟qƴ;Gv(~feǷgmǷ)*PforjY=1;:::;h.F*romthefactthatLF (tj6mij)=tj~feǷgm cijCwe3conclude3degŸqƴ;G 2(tjmij)=degnqƴ;Gy޲(~feǷgmǷ).3This,inturn,ʍimpliesUULFt(Pލ ;h% ;jg=1cj6tjmij)=Pލ USh% USjg=1Vcj6tj~feǷgm cij(=~feǷgm ϲ,UUwhichconcludestheproGof.X Dr"궲Ifzhwewantto ndallelementsofzhSyz(GѲ)zhusingthisproGcessoflifting,weJneedtoascertainthatthereexistsasystemofgeneratorsofJSyz(GѲ)Jcon-sistingTofliftings.ThisisachievedTbythefollowingpropGositionwhoseproofdemonstratesUUoncemorethepGowerUUoftermorderings.PropQositionT2.3.11.wL}'et?f~feǷgmǷ1 D*;:::;~feǷgm r_t gbe?ahomogeneoussystemofgener-atorsofthemo}'duleSyz:(LMjC(GѲ)),andletm1|s;:::;mtLn2Syz7(G)b}'eelementssuchthatLF'(miTL)4=~feǷgm i fori=1;:::;t.Thenfm1|s;:::;mtVgisasystemofgener}'atorsofSyz(GѲ).Pr}'oof.6F*orFcontradictionweassumethatthesubsetFSӲofSyz3e(GѲ)ofFsyzygieswhicharenotgeneratedbyfm1|s;:::;mtVgisnotempty*.Bythefundamen-tal`propGertyoftermorderings(seeTheorem1.4.19),thereexists`mҸ2S궲with#minimal#degFqƴ;G!.ThenthereexistsanaturalnumbGer#hsuch#thatweʍhave2YLFx(m)7r=Pލŭh%ŭjg=1cj6tj~feǷgm cijAiwith2Ycjn27rKSn7f0g,2Ywithtj27rT^nq~,2Yandwith@궵ijĸ2f1;:::;tg֪forjY=1;:::;h.֪Theelement֪m^0Q=m;Pލ ƴh% i=1轵ciTLtimij Wsatis eseither‹m^0KU=}0ordegqƴ;G q(m^09)<߲degAqƴ;G$(m).‹InbGothcaseswegetacontra-diction,UUandtheproGofiscomplete.zr"궲The@' nalpropGositioninthissectionisthegemwepromisedintheintro-duction.k7Wō>; O2.3Syzygies9107-ō>;PropQositionT2.3.12.wL}'etS[g1|s;:::;gs R2Pc^r}nf0gandM3=hg1|s;:::;gsF:i.S[Then ConditionsVA1|s),A2)ofVPr}'oposition2.1.1andConditionsVD1|s),D2),D3)ofPr}'oposition2.3.10ar}'eequivalent.Pr}'oof.6FirstZweshowthatConditionZA2|s)impliesD1).ZLetmc=Pލ ^s% ^jg=1_fj6"j궲bGeNanon-zerohomogeneouselementofNSyz(LMjC(GѲ)).W*emaysuppGosethat궵(m)ڸ6=0,0sinceincase0(m)=0we0havemڸ2Syz(GѲ)andLFO(m)=m,i.e.{degDqƴ;G$*(h)andLFM(mYh)=LF>(m)=m.ThusUUtheelementUUm8hisUUaliftingofm."Nowletusshowthereverseimplication.W*eassumeforcontradictionthatthere]existsanelement]v"2M nf0gwhich]cannotbGerepresentedasrequestedbybConditionbA2|s).W*eobservethatifbv簲=ןPލs%i=19 fiTLgiforsomepGolynomials궵f1|s;:::;fs›2|aPDand[if[m=Pލ s% jg=1 fj6"j6,thenwehave[v:=|a(m).Inotherwords,itheelementimisiapreimageofiv=Bunder.iBythefundamentalprop-erty:=oftermorderings(seeTheorem1.4.19),weknowthatamongallpreim-agesofvWunder,thereexistsonepreimagemwithminimaldegvqƴ;G (m).W*e[cannothave[degqƴ;G >Z(m)'=LT[-(v[ٲ),[bGecauseotherwisetherepresentation궵v=[؟Pލ s% i=1 fiTLgiisalreadyoftheformrequiredbyConditionA2|s).ThereforePropGosition2.3.6.ashowsthatwemusthaveL*T!(v[ٲ)< idegqƴ;G%(m).Next,PropGosition,2.3.6.byields,LFK(m)2Syz7(LMjC(GѲ)).,ThusCondition,D1|s)givesusJanelementJѵm^0.r=`9Pލts%tjg=1˵f^0;Zj6"j2`9SyzX(GѲ)suchJthatLF(m^09)`9=LF'X(m).Inparticular,xzcg..Byadding%suitable0 ro9wstothismatrix,showhowonecanproAducenon-trivialsyzygies%ofTthetripleTG=(x-=288yR;xy`zc;yR-=2xzc).%Exercise 3.ǩInBthecaseBnA=2,P%$=Q[x;yR],r=2,Bcomputeasystem%ofvgeneratorsofthesyzygymoAduleofthetuplevG=((xy%+.{yR;x);(xyR;y);%(x;x8+yR);(x;y))Tb9yThand.%Exercise4.ۏLetƗP=6K[x;yR;zc]bAeƗapolynomialringo9verƗa eldƗK,%letrr=R1,andletG٢=R(x;yR;zc).ComputethesyzygymoAduleofasetof%generatorsTofTSyz7?P^(G).lNWō>;1084`2.Gr`obnerTBases-ō>;%Exerciseo5. oGiv9e_VadirectproAofforthefactthatCondition_VD1*)of %PropAositionT2.3.10impliesConditionTB2*)ofTProposition2.1.2.%Hint:»If»m2Manif0ghas»aleadingtermoutsideN,pic9kapreimageofm%underTofTsmallestR-degreeandloAokatthefundamen9taldiagram.%Exercise 6.LetA(g1*;::: ;gsܙ2PH-=rn+pf0g,A(letMӶ=hg1*;::: ;gsi,A(andletA(G%bAeTthetupleT(g1*;::: ;gs). *?a)7xPro9vethatSyz{(G)=0ifandonlyifMisafreePH-moAdulewithbasis7xfg1*;::: ;gsg.)b)7xLet>s=3,>letn=3,>letr=2,>letg1=(x-=2*;x)yR),>letg2=(0;y),7xandTletTg3m=(xyR;zc).Thensho9wthatSyz7(G)6=0.%ExerciseR#7.@Letg1*;::: ;gs2PH-=r~n;+f0g,letM=hg1*;::: ;gsi,letG2bAethe%s-tuple|(g1*;::: ;gs),|letTbAeamoduletermorderingon|T-=n7he1*;::: ;er,pi,%andletL:T̟(gi,r)=tie Di withti82T-=n Sand i82f1;::: ;rAgfori=1;::: ;s. *?a)7xPro9veTthatTM8isafreePH-moAduleif i86= j/foralli6=j.)b)7xDeduceDQthatthesubmoAduleofDQPH-=3 generatedb9ythesetofvectors7xf(x;y`8zc;x);(z;yR-=28x;x);(z-=2l8y`+1;yR-=2x;x3)gTisTfree.)pT utorialT19:SyzygiesofElemen9tsofMonomialMoQdules@궲Let9KUbGe9a eld,nd1,Pt=K[x1|s;:::;xnq~],r1,9andM|Pc^r ja monomial(submoGdulegeneratedby(ft1|se 1u;:::;tsF:e sg,wheret1|s;:::;ts]2T^n궲andUU 1|s;:::; s R2f1;:::;rGg.a)%UseTheorem2.3.7togiveanexplicitsystemofgeneratorsofthesyzygy%moGduleof(t1|se 1u;:::;tsF:e s).W*riteaC0oLCoA"functionMonomialSyz>d(::: )%whichtakesasystemofgeneratorsofamonomialmoGduleM-asabove%andUUcomputesits rstsyzygymoGdule.?b)%ShowMbyexamplethatthesystemofgeneratorsofthesyzygymoGdule%givenmina)isingeneralnotminimal,evenifmft1|se 1u;:::;tsF:e sgismminimal.\Ic)%ApplyyourfunctionMonomialSyz>ZԲ(::: )tocomputethesyzygymoGdules%ofUUthefollowingtuples.)1)8x(x^34xy[ٟ^7L;x^23y[ٟ^19Կ)Q[x;y[ٲ]^2)2)8x(x;y[;zp)Q[x;y;zp]^3)3)8x(xy[;yzp;xz)Q[x;y[;zp]^3)4)8x(xe1|s;y[e1;y[e2;zpe2;xe3;zpe3)(Q[x;y[;zp]^3|s)^6?ֲd)%Show:thatif:r^=1,1ic).Eachtime,trytodeterminewhetherthecomputedsystemofgen-%eratorsUUofthesyzygymoGduleisminimal.mfJWō>; O2.3Syzygies9109-ō>;T utorialT20:LiftingofSyzygies궲In8thistutorialweshalltrytoprogramtheliftingofsyzygiesdiscussedin the;llastpartofthecurrentsection.Asusual,let;lKbGea eld,letn1,;llet궵Pi=`K[x1|s;:::;xnq~]{MbGe{Mapolynomialring,let{MrM}`1,let&bGeamoduletermorderingEonET^nq~he1|s;:::;ermi,letG^=(g1|s;:::;gsF:)2(Pc^r1)^sbGeEatupleofnon-zerovectors,yandletyMn=Shg1|s;:::;gsF:iPc^r1.yW*eassumethatConditionsD1|s),궵D2|s),]and]D3)are]satis ed.F*oriz=1;:::;s,]wewriteL*T*(giTL)z=tie i ewith궵tix=2#T^n and 1 irG, and,for i;j|2#f1;:::;sgsuch that ix==# j6,welet궵tij =lcmUS(tiTL;tj6)=tid=tj=gcd(tiTL;tj).a)%Show/that,for/1ik=3sij ;1Pލ ?s% ?k+B=1uzfijgk v"j 9are%liftingsUUofUUij `Mforalli;jasabGove.\Ic)%ConcludeLthatthesetLfsij n_jcg1iandin-%dices漵i;jyGasabGoveandcomputesalistofpGolynomials[fijg1 k;:::;fijgs 2]%correspGondingUUtotherepresentationina).\Ie)%UsingtheprogramMonomialSyz=s(::: )fromT*utorial19andStdRepr(sDz(::: )%as3usubfunctions,writeaC0oLCoA#program3uLiftSyz)`(::: )3uwhichtakesthe%tupleUUG&andUUcomputesthelistofallsyzygiessij `Masinb).f)%Using~themoGduletermorderingDegRevLexPos,computethelistsofall%syzygiesUUij `MandUUallsijinUUthefollowingcases.)1)8xG^=(x^2l1S8x2|s;x^2l2x3|s;x^2l3x1|s)2Q[x1;x2;x3]^3)2)8xG^=(x1|se1;x2e1;x3e2;x1e3)2(Q[x1|s;x2;x3]^3)^4)3)8xG^=(x1|sx4x2x3;x1x^2l3x^2l2x4;x^2l1x3x^3l2;x2x^2l4x^3l3)2Q[x1;x2;x3;x4]^4nzWō>;1104`2.Gr`obnerTBases-ō>;2.4**Gr@obnerBasesofIdealsandMo`dulesTheNXofyagivensubmoGduleyM,i.e.Faminimalsub eldofFKvbwhichcontainsthecoGecientsofsomesystemofUUgeneratorsofUUM.owWō>;{2.4Gr`obnerTBasesofIdealsandMoAdules9111-ō>;"궲NowA?westartthemainpartofthissectionbyrecallingthat,asusual, weJletJKfbGea eld,n1,PL=K[x1|s;:::;xnq~]aJpGolynomialring,r0"1,andE)aEmoGduletermorderingonT^nq~he1|s;:::;ermi.InthefollowingtheoremweUUcollectalltheconditionsstudiedintheprevioussections.TheoremT2.4.1.dH(CharacterizationTofGr@obnerBases)F;or&asetofelements&G/=fg1|s;:::;gsF:gPc^rXnPf0gwhich&gener}'atesasub-󻍑mo}'dulebCMT =;1124`2.Gr`obnerTBases-ō>;2.4.A ExistenceTofGr@obnerBases궲Our|U rsttaskistoshowtheexistenceofGrobnerbases.IfwerecallPropGo- sition6f1.5.6.b,itisclearthatthereareelements6fg1|s;:::;gsm2>3MMsatisfyingConditionB2|s).ButdotheygenerateM?OurnextpropGositionanswersthisquestionUUarmatively*.PropQositionT2.4.3.q(ExistenceTofaT[-Gr@obnerBasis)~L}'etMbeanon-zer}'oPc-submoduleofPc^r1.~oa)%GivenSg1|s;:::;gs R2MJn/f0gsuchSthatL*Tx(M)=hL*T %(g1|s);:::;L*TN7Ͳ(gsF:)i,%wehaveM=hg1|s;:::;gsF:i,andthesetG=fg1|s;:::;gsF:gisa[-Grobner%b}'asisofM.Hb)%Themo}'duleMhasa[-Grobnerb}'asisG=fg1|s;:::;gsF:gMOn8f0g.Pr}'oof.6Firstqweshowclaima)bycontradiction.SuppGoseqhg1|s;:::;gsF:i^M.ByŒTheorem1.4.19,thereexistsanelementŒm,и2MExn.]hg1|s;:::;gsF:iwhoseleading)term)L*Tu0 (m))isminimalwithrespGecttoamongallelementsofthat|set.Sincewehave|L*T t (m) 2L*T,(M)=hL*T %(g1|s);:::;L*TN7Ͳ(gsF:)i,|thereareYc2Ktnf0g,t2T^nq~,Yandi2f1;:::;sgsuchYthatLMd(m)=ctLMUw(giTL).ThuswegetL*T$ (mbctgiTL)<)L*T=/Ӳ(m),andhencembctgid2hg1|s;:::;gsF:i,contradictingUUm=2 8hg1|s;:::;gsF:i."ClaimUUb)followsfroma)usingPropGosition1.5.6.b.V"궲TheexistenceofGrobnerbasesimpliesoneofthemostimpGortantproper-tiesP*ofpGolynomialringsoverP* elds.InSection1.3wedescribedthepropertyofkbGeingNoetherianinthecaseofmonoideals.Usingasimilarformulation,weUUextendittoidealsandmoGdules.De nitionT2.4.4.is~ABringX(resp.moGdule)iscalledNoQetherianifeveryas-cendingUUchainofideals(resp.submoGdules)becomeseventuallyUUstationary*."TheefollowingcharacterizationsofNoGetherianmodulesareincompleteanalogywiththecaseofNoGetherianmonoidsandcanbeshownexactlyasPropGositionUU1.3.4.PropQositionT2.4.5.qL}'et>RQbe>aringandMU0anR-mo}'dule.Thefollowingc}'onditionsareequivalent.~oa)%Everysubmo}'duleofMis nitelygenerated.Hb)%Everyasc}'endingchainN1; ȵN2m&ofsubmo}'dulesofMiseventually%stationary.Hc)%Every3Vnon-emptysetofsubmo}'dulesofM3-hasamaximalelement(with%r}'especttoinclusion)."궲As(BaconsequenceofPropGosition2.4.3,weobtainaversionofHilbGert'sBasisTheoremfor nitelygeneratedmoGdulesover nitelygeneratedK-algebras.q8Wō>;{2.4Gr`obnerTBasesofIdealsandMoAdules9113-ō>;TheoremT2.4.6.dH(HilbQert'sTBasisTheorem) Everyd nitelygener}'ateddmoduleovera nitelygenerateddK-algebraisNo}'etherian.Inparticular,P*=K[x1|s;:::;xnq~]isaNoetherianring.Pr}'oof.6IfwerepresenttheK-algebraintheformPV=I吲withapGolynomialringrP[=[K[x1|s;:::;xnq~]andranidealrI=Pc,wecanviewthemoGdulerMasa+ nitelygenerated+ƵPc-moGduleviathecanonicalmapP* GPV=I.Obviouslyit+sucestoshowthatevery+Pc-submoGduleofMB)is nitelygenerated.Since궵M'is nitelygenerated,wecanrepresent M'intheformM=Pc^r1=U'with궵r׸1gandgasubmoGduleUոPc^r1.SinceeverysubmoGduleofMisoftheform'sNA=U>with'sasubmoGduleN3Pc^r1,itsucestoshowthatevery'sPc-sub-moGdule\of\Pc^r .is nitelygenerated,andthisisanimmediateconsequenceofPropGositionUU2.4.3.%2.4.B NormalTF orms궲OurnextapplicationofGrobnerbasesistoshowhowtheyhelpustopGerforme ectivecalculationsinaresidueclassmoGdule댵Pc^r1=qM.Severalattemptstosolvethisquestionhavefailedsofar,bGecausewewerenotableto ndauniquerepresentativeinPc^r EforaresidueclassinPc^r1=qM.UsingaGrobnerbasis,UUwenow ndthatallthoseattemptsleadtothesameuniqueanswer."LetG=fg1|s;:::;gsF:gPc^r nf0gbGea[ٲ-GrobnerbasisofM Ȳ=궸hg1|s;:::;gsF:imPc^r&,nandletnmm2Pc^r1.nByConditionC3|s),thereexistsaunique󻍑elementwmG 2Pc^r suchwthatmT΍G2!qǵmG QandsuchthatmG Qisirreduciblewith 󺍑respGect\StoT΍ MG2\S x!xʲ.A\QpriorithiselementseemstodependontheGrobnerbasischosen,UUbutindeeditdoGesnot,asthefollowingpropGositionshows.PropQositionT2.4.7.qIn 'theab}'ovesituation, 'mG istheuniqueelementofPc^rwithBthepr}'opertiesBthatBmmG 2صMYandSupp{(mGڲ)\L*T["fMgز=;.BInp}'articular,itdoesnotdependontheparticular[-Grobnerbasischosen.Pr}'oof.6W*e1knowthat1m_smG a2'MLand1thatthesuppGortofmG I doGesnotintersectL*T5@fMg.Uniquenessfollowsfromtheobservqationthat,fortwosuchelementsmG WandmH,thesuppGortofmGmH \ɸ2MdoGesnotintersectL*T E%۸fMg,_andthisis,byCondition_C2|s),onlypGossibleif_mGFиmH \ɲ=0. De nitionT2.4.8.is~LetM3Pc^r bGeanon-zeromodule,andletm2Pc^r1.Theelement mG L2rPc^r $describGed above iscalledthenormal|formof mwithrespGectto[ٲ.ItisdenotedbyNFŸ;M!{(m),orsimplybyNFŸ{X(m)ifitisclearwhichUUsubmoGduleisconsidered."Below wecollectsomepropGertiesofnormalforms.Inparticular,weseethattheDivisionAlgorithmwithrespGecttoaGrobnerbasisprovidesane ectiveUUmethoGdforcomputingnormalforms.rĵWō>;1144`2.Gr`obnerTBases-ō>;CorollaryT2.4.9.fIntheab}'ovesituation,letG^=(g1|s;:::;gsF:).~oa)%Ifwm2Pc^r&,wthenNRc;GZ(m)agr}'eeswwithNF,(m).Inp}'articular,thenor- %malr}'emainderdoesnotdependontheorderoftheelementsg1|s;:::;gsF:.Hb)%F;orm1|s;m2C2Pc^r&,wehaveNF(m1S8m2|s)=NF70Ͳ(m1)8NF?(m2).Hc)%F;orm2Pc^r&,wehaveNF(NFi(m))=NF70Ͳ(m).Pr}'oof.6Claima)followsfrom۵mNRa;GUѲ(m)2MandfromthefactthatthesuppGortdofdNRן;G SG(m)ddoesnotmeetdL*TfMg.Nextweshowb).W*ehave궵m1u!m2(NFi(m1|s)NF͟bc(m2))=(m1u!NF͟bc(m1|s))(m2NF͟bc(m2|s))2M󻍑궲andNFO(m1|s)ZNFa>Բ(m2)isirreduciblewithrespGecttoT΍CG2 j!k.Theuniquenessofpsuchanelementyieldstheconclusion.Claimc)followssimilarly*,bGecauseNF!՟'Tk(m)NFgD(m)=02MƲandNFʟ!`(m)isirreduciblewithrespGecttoT΍G2 !!".X D"궲F*or5thepurpGosesofactualcomputations,oneofthemostusefulappli-cationsofnormalformsisthepGossibilitytocheckwhetheranelementiscontainedinasubmoGduleorwhetheronesubmoduleiscontainedinanother.PropQositionT2.4.10.w(SubmoQduleTMem9bershipT est)L}'etfg1|s;:::;gsF:gPc^r &generateaPc-submoduleM3=hg1|s;:::;gsF:iofPc^r&,andletfh1|s;:::;htVgPc^r egener}'ateaPc-submoduleN3=hh1|s;:::;htViPc^r1.~oa)%F;orFm1|s;m2ф2UPc^r&,Fwehavem1qm2ф2UMaifandonlyifNFe;M"(m1|s)U=%NF3;MCmw(m2|s)._Inp}'articular,anelement_m=2Pc^r satis esm2Mzif_and%onlyifNF;M"O(m)=0.Hb)%WehaveN3MifandonlyifNF;M"O(hiTL)=0fori=1;:::;t.Hc)%Thec}'onditionMѲ=춵NisequivalenttoNFʟ;N!ز(giTL)=NF՟;M (hj6)=0for%i=1;:::;sandjY=1;:::;t.~od)%IfN3MandL*T7v fMgL*Tj=fNg,thenM3=N.Pr}'oof.6T*oshowthe rstclaim,letm1|s;m22IPc^r tͲsuchthatm1邸mm22IM.Then;0=NF7;M (m1 "m2|s)=NF7;M(m1|s)NFΟ;MH(m2);by;Corollary2.4.9.b.Conversely*,letNF4;M#B(m1|s)ī=NFʟ;M"(m2).Inthiscase,theclaimfollowsfromUUm1S8m2C=(m1NF?;M(m1|s))(m2NF?;M(m2|s))2M."Clearly*,jEclaimb)isaconsequenceofa),andclaimc)followsfromb).Thusitremainstoproveclaimd).SincewehaveNM,itisclearthatL*T E%۸fNg>L*TYDfMg.&ThusthehypGothesisthatwehavetheotherinclusionL*T E%۸fMgL*T_fNgimpliesequalityL*T%(fNg=L*T_fMg.Nowtakeanelementȵm2M.W*ehaveSupp(NF;Nq-(m))׸\L*T=ffNg=;,andthereforeSupp*#(NF;Nq-(m))6\L*T8ş[fMg=;.TheuniquenessinPropGosition2.4.7showsthatUUNF\t;N Ƃ(m)=0,UUi.e.wegetUUm2N.m"궲AsanimpGortantapplicationofthenotionofGrobnerbasis,wegetanewversionUUofMacaulay'sBasisTheorem1.5.7.s։Wō>;{2.4Gr`obnerTBasesofIdealsandMoAdules9115-ō>;CorollaryT2.4.11.l(NewTV ersionofMacaula9y'sBasisTheorem) L}'etMhjPc^r beaPc-submodule,letGhj=fg1|s;:::;gsF:gPc^r enF4f0gb}'ea궵[-GrobnerGb}'asisofM,andletBb}'ethesetofalltermsinT^nq~he1|s;:::;ermiwhichFar}'enotamultipleofanyterminthesetFfL*T %(g1|s);:::;L*TN7Ͳ(gsF:)g.Thenther}'esidueclassesoftheelementsofBXformaK-b}'asisofPc^r1=qM.bPr}'oof.6ThefactthatGisa[ٲ-GrobnerbasisofMimpliesthatL*Tx'fMg궲isgeneratedbyfL*T %(g1|s);:::;L*TN7Ͳ(gsF:)gbyConditionB1|s)ofTheorem2.4.1.SoUUthestatementfollowsimmediatelyfromTheorem1.5.7.C5$~V2.4.C ReducedTGr@obnerBases荑궲InthelastpartofthissectionweaddressthequestionofuniquenessofGrobnerwbasesandprovideanapplicationofit.GivenamoGduletermor-deringC[ٲ,CasubmoGduleMPc^r thasmany[ٲ-Grobnerbases.F*orinstance,wemcanaddarbitraryelementsofmM/toa[ٲ-Grobnerbasisanditremainsas;[ٲ-Grobners;basisofM.However,s;thereisauniqueonewhichsatis esthefollowingUUadditionalconditions.De nitionT2.4.12.o3|LetTG=fg1|s;:::;gsF:gPc^rܸn7f0gandM3=hg1|s;:::;gsF:i.W*es0saythats0Giss0areduced[-Gr@obnerbasisofs0MKifthefollowingcon-ditionsUUaresatis ed.a)%F*orUUi=1;:::;s,UUwehaveUULC:0в(giTL)=1.?b)%The˰set˰fL*T %(g1|s);:::;L*TN7Ͳ(gsF:)gisaminimalsystemofgeneratorsof%L*T2U.7IJ(M).\Ic)%F*orUUi=1;:::;s,UUwehaveUUSupp=(gi,8L*To?(giTL))8\L*TofMg=;.TheoremT2.4.13.jUF(ExistenceŠandUniquenessofReducedGr@obnerBases)F;or revery rPc-submo}'duleMθųPc^r&,ther}'eexistsauniquereduced r[-Grobnerb}'asis.Pr}'oof.6W*e)startbyprovingexistence.Let)Gk=fg1|s;:::;gsF:gbGe)any[ٲ-Grobner]basisof]M.IfwereplacegiRbyLCwBز(giTL)^1 tgiforiβ=1;:::;s,]weobtainaGrobnerbasiswithpropGertya).ByConditionB2|s)ofTheorem2.4.1,themonomialmoGduleL*T?e(M)isgeneratedbyfL*T %(g1|s);:::;L*TN7Ͳ(gsF:)g.ThenweusePropGosition1.3.11.btogetfromthissettheuniqueminimalsys-temm2ofgeneratorsofm2L*TsW(M).AfterpGossiblyrenumberingm2thevectorswemayassumewthatthisminimalsystemofgeneratorsiswfL*T %(g1|s);:::;L*TN7Ͳ(gtV)g,whereC/tSs.C/AndusingagainConditionB2|s)andPropGosition2.4.3.a,weseeLkthatthesetLkG^0Q=fg1|s;:::;gtVgisLka[ٲ-GrobnerbasisofMcwhichsatis esconditionsUUa)andb)ofthede nition."Now@wewrite@giҕ=~IL*T!؟n(giTL)'+hiTL,andifwelet@g^[ٷ0;Ziҕ=~IL*T!؟n(giTL)+NFFܲ(hi)forRig=1;:::;t,RwecanformthesetG^00ٲ=gfg^[ٷ0l1|s;:::;g^[ٷ0፴tVg.W*eclaimthatG^00궲islareducedl[ٲ-GrobnerbasisofM.Sinceg^[ٷ0;ZiAF=giSH(hiNFO&(hiTL)),lweusetWō>;1164`2.Gr`obnerTBases-ō>;궲PropGositionh2.4.7andgethg^[ٷ0;Zi;ظ2猵Mfori=1;:::;t.hByConditionhB2|s),the setWڵG^00 tLisWa[ٲ-GrobnerbasisofM.Sinceitclearlysatis esconditionsa)andb)ofthede nition,itremainstoprovethatitalsosatis esconditionc).Indeed,oforeveryoi2f1;:::;tg,onoterminSuppRW(NFi(hiTL))liesinL*TfMg,ݍbGecauseUUNF\t (hiTL)UUisUUirreduciblewithrespecttoT΍jGr02UU q!q̲."Finally*,toshowuniqueness,weassumethatG=fg1|s;:::;gsF:gand궵H=fh1|s;:::;htVg2are2tworeduced2[ٲ-GrobnerbasesofM.F*romthefactthattherminimalmonomialsystemofgeneratorsofamonomialmoGduleisunique(seeIePropGosition1.3.11.b),weconcludeIes=tandIethatwecanrenumbGertheelementsuofuHassuchthatuL*T5(giTL)+M=L*Tܟ1r(hi)ufori+M=1;:::;s.uMoreover,for궵i=1;:::;s,wehavegiu)hid2M,andgiu)higEis,byconditionc)ofthede -󻍑nition,-irreduciblewithrespGecttoT΍ |G2- +!+.Thusproperty-C2|s)of-Theorem2.4.1provesUUgid=hifori=1;:::;s.1W"궲As\anapplicationoftheexistenceanduniquenessofreduced\[ٲ-Grobnerbaseswecanshowtheexistenceanduniquenessofa eldofde nitionforsubmoGdulesUUofUUPc^r1.mDe nitionT2.4.14.o3|LetKbGea eld,P*=K[x1|s;:::;xnq~]apGolynomialring,andUUM3Pc^r &aPc-submoGdule.a)%Letk\ KbGeasub eld.W*esaythatM/isde ned/o9verkhifthere%existUUelementsinUUkP[x1|s;:::;xnq~]^rwhichgenerateUUMlpasaPc-moGdule.?b)%A sub eldkN4KN7iscalleda eld ofde nitionofM6ifMisde ned%overּk'SandּthereexistsnopropGersub eldkP^0k'SsuchthatMײisde ned%overUUkP^0в."It{isclearthatifa eldofde nitionofa{Pc-submoGduleMS<Pc^r lexists,itUUhastocontaintheprime eldofUUK.LetusloGokataconcreteexample.ExampleT2.4.15.h`Let9sI *CHC[x1|s;x2;x3]9sbGe9stheidealgeneratedbytheset궸fx^2l1iPp PfeE5x1|sx2+i3x1x3+i2PpUWPfeE5 UXx^2l3;I x1x2iPp PfeE2x^2l3;I 2x1x2+iPp PfeE3x^2l3g.aObviously*,theUUidealUUI7isde nedoverQ[PpUWPfeE2 UX;Pp PfeE3;Pp PfeE5]."ButtitisalsoeasytochecktthattɵIf=(x^2l1K+M3x1|sx3;qx1x2;x^2l3).tTherefore,theuidealuIWisde nedoverutheprime eldQofC,andtheunique eldofde nitionUUofUUI7isQ."TheBfollowinglemmacapturesoneimpGortantaspGectoftheproofoftheexistenceUUanduniquenessofthe eldofde nition.LemmaT2.4.16.bG L}'etK^0.K4bea eldextension,letPc^0`{=.K^0U[x1|s;:::;xnq~],lettdM^0l(Pc^01Ȳ)^rb}'etdaPc^01-submoduletdof(Pc^01Ȳ)^rm,andletMb}'ethePc-submo}'duleofPc^r egener}'atedbytheelementsofM^0T.~oa)%A k[-Grobner b}'asisof M^0isalsoa[-Grobnerb}'asisofM.Inp}'articular,%wehaveL*T7v fM^0Tg=L*Tj=fMg.Hb)%Ther1r}'educedr1[-Grobnerbasisofr1M^0Wisalsothereducedr1[-Grobnerbasis%ofM.u Wō>;{2.4Gr`obnerTBasesofIdealsandMoAdules9117-ō>;Pr}'oof.6LetXkGv=fg1|s;:::;gsF:g(Pc^01Ȳ)^rS:nf0gbGeXka[ٲ-GrobnerbasisofM^0T. Since[theset[Ggenerates[thePc^01Ȳ-moGduleM^0AIandthesetM^0AIgeneratesthe궵Pc-moGduleUUM,UUthesetGgeneratesthePc-moGduleM."LetGYbGethetuple(g1|s;:::;gsF:).UsingTheorem2.3.7,weseethatSyz#1(L*T %(GѲ))=hij j1imentsb ij mhaveb liftingsin(Pc^01Ȳ)^sF:.TheseliftingsarealsoliftingsinPc^s Բoftheelements&ij if&weconsiderthoseaselementsof&Pc^sɲ.UsingConditionD3|s)ofTheoremW2.4.1,wededucethatWGisWinfactaW[ٲ-GrobnerbasisofM.ThisprovesUUa)."T*oproveb),weobservethattheextraconditionsrequiredinDe ni-tionUU2.4.12areindepGendentofthebase eld.P=׍TheoremT2.4.17.jUF(ExistenceandUniquenessoftheFieldofDe ni-tion)L}'etMbeanon-zer}'oPc-submoduleofPc^r1.~oa)%Ther}'eexistsaunique eldofde nitionofM.Hb)%Givenanymo}'duletermordering[,letGb}'ethecorrespondingreduced%[-Grobnerb}'asisofM.Thenthe eldofde nitionofM#isthe eld%gener}'atedxovertheprime eldofxK˔bythec}'oecientsxofthetermsinthe%supp}'ortofthevectorsinG.Pr}'oof.6Let bGeamoduletermordering,andletGbethereduced[ٲ-GrobnerbasisofM.Moreover,letkbGethe eldgeneratedovertheprime eldofҵKbythecoGecientsoftheelementsofҵG.SincethesetGgener-atesUUM,UUthemoGduleMlpisde nedoverkP."SuppGoseKnowthatKK^0LmKgisasub eldoverwhichKMfisde ned,i.e.sup-pGoseYthereexistsasystemofgeneratorsYfm1|s;:::;mtVgofYthePc-moduleM궲which/ iscontainedin/ K^0U[x1|s;:::;xnq~]^rnXf0g.LetG^0 U}=Dfg^[ٷ0l1|s;:::;g^[ٷ0፴sF:g궵K^0U[x1|s;:::;xnq~]^r|bGe_thereduced_ڵ[ٲ-GrobnerbasisoftheK^0U[x1|s;:::;xnq~]-moGdule궸hm1|s;:::;mtViK^0U[x1;:::;xnq~]^rm.tVSincethereducedtV[ٲ-GrobnerbasisofamoGd-uleisunique,Lemma2.4.16.bimpliesµGj˲=G^09.F*romthisweinferthat궵kK^0U."ThefactsthatM/isde nedoverkP,andthateveryother eldoverwhichM2isde nedcontainskP,togetherimplybGothclaimsofthetheo-rem.01Z%Exerciseuz1.vLet%I=hx(gR)withg"2Pndf0gbAe%aprincipalidealin%PH. %Sho9wthatG=fgRgisaGr`obnerbasisofIfwithrespAecttoev9eryterm%ordering.偍%ExerciseC2.vLet(m1*;::: ;ms2PH-=r {bAe(terms,andletM=hm1*;::: ;msi.%Sho9w2that2fm1*;::: ;msgisaGr`obnerbasisofMwithrespAecttoev9ery%termTordering.v Wō>;1184`2.Gr`obnerTBases-ō>;%Exercise$d3.LetTG=fg1*;::: ;gsgPH-=rn8f0gbAeTaR-Gr`obnerbasisof %the?PH-moAduleMт=؞hg1*;::: ;gsi,?andlet?m2M.Sho9wthat?G*[fmgis%aTR-Gr`obnerTbasisofM.%Exercise4.hLet pg1T=*Ix2Gx-=2h1 5andg2=*Ix3Gx-=3h1 5bAe ppolynomialsin%K[x1*;x2;x3].OFindatermorderingOjonT-=3zgsuc9hOthatG%=fg1*;g2gOisOa%R-Gr`obner basisoftheideal IF!=(g1*;g2), andatermordering suc9hthat%itTisnot.%Exercise85.GLet8Pک=K[x1*;::: ;xn7],8letmn,8letG=ff1*;f2;::: ;fmg,%whereAfi2ۏK[xi,r]fori=1;::: ;m,AandletAIP|bAetheidealgenerated%b9yTG. *?a)7xUse Condition C3*)ofTheorem2.4.1tosho9wthatGisaR-Gr`obner7xbasisTofTIɯwithrespAecttoev9erytermorderingR.)b)7xIf,1moreo9ver,thepAolynomials1fiBaremonic,showthat1Gisthereduced7xR-Gr`obnerTbasisofTIɯwithrespAecttoev9erytermorderingR.%Exercise6.VLetRbAeaNoetherianin9tegraldomain.Showthatthe%follo9wingTconditionsareequiv|ralent. *?a)7xF:orTallTa;b2Rvn8f0g,theidealT(a)\(b)isTprincipal.)b)7xTheTringTR%isfactorial.%Hint:TUseExercise6inSection1.2.%Exercise+*7.MUsinguCorollary2.4.11andC'oZ:CoA, ndasetoftermswhose%residueF+classesformabasisofF+Z=(5)[x;yR;zc]=(x-=2oyz;y-=37+oz-=34;z-=5x-=2*yR-=2}Q)F+asF+a%Z=(5)-v9ectorspace.(Hint:Y:oumayusetheC'oZ:CoAfunctionGBasis(...).)%Exercise8.XAsystemofgeneratorsG=fg1*;::: ;gsgPH-=r!nf0gof%aPH-moAduleM>A=E]hg1*;::: ;gsiPH-=r iscalledaminimalR-Grobner%basisXofMiffL:T k(g1*);::: ;L:T ;Z?(gs)gisaminimalsystemofgenerators%ofTL:Tɿ(M). *?a)7xPro9ve"thatan9ytwominimal"R-Gr`obnerbasesofMha9ve"thesame7xn9umbAerTofelemen9ts.)b)7xGiv9eFanexampleofamoAduleFM?whichhastwodi erentminimal7xR-Gr`obnerbases,allofwhoseelemen9tsgi/haveleadingcoAecien9ts7xLCCHH(gi,r)=1.%Exercise9. Let=gbAe=gatermorderingonT-=n7he1*;::: ;er,pi.W:eset%L:T1bO6f (0)=1.Inparticular,w9eareassumingL:TQ (gR)<L:TF1I(0)forevery%g^2 4PH-=ruS.^Giv9enatuple^(g1*;::: ;gs)2(PH-=ruS)-=s,w9eidentifyitwiththetuple%(g1*;::: ;gs;0)2(PH-=ruS)-=s+1H,&hencewith&(g1;::: ;gs;0;0)2(PH-=ruS)-=s+2H,&andso%on.%F:orm=t9wotuplesm=GA=$J(g1*;::: ;gs)2(PH-=ruS)-=soandG-=0]=(g-=R0h1*;::: ;g-=R0 sG0:)2(PH-=ruS)-=s{q0:,%w9ede neGYkbG-=0УifandonlyifL:THLP(G)kbLexL:TEH I(G-=08).Thismeansthat%either6thereexistsanindex6it1suc9h6thatL:Ti$(gi,r)< wL:T(,?(g-=R0~ƍi)6and%L:T1bO6f (gj)=L:TBvF1(g-=R0~ƍj)Tfor1j;{2.4Gr`obnerTBasesofIdealsandMoAdules9119-ō>;)b)7xPro9ve thattherelation is re exiv9eandtransitive,butnotato- 7xtalUHorderingonthesetoftheincreasinglyorderedtuplesofelemen9ts7xofTPH-=ruS.*6c)7xLetMdbAeanon-zerosubmoduleofPH-=ruS,andletGwbAeanincreasingly7xordered,win9terreducedtupleofelementsofwM.Showthatthefollowing7xconditionsTareequiv|ralen9t. <1)I?qWithrespAectto,thetupleG isminimalamongallincreasinglyI?qordered,Tin9terreduced,monictuplesofelementsofTM.<2)I?qThe Ktuple KGBisobtainedb9yincreasinglyorderingthereducedR-I?qGr`obnerTbasisofTM.%Exercise110.Let-YIbAe-YanidealofP=K[x1*;::: ;xn7],letbAeaterm%orderingjonjT-=n7,andletMbAeagroupofK-algebraautomorphismsofPH.%Sho9wbyexamplethatifIjisH-stable(i.e.if (I[)+Ijforall  2H),%thenTthereducedTR-Gr`obnerbasisofIɯneednotbAeH-stable.=T utorialT21:LinearAlgebra궲TheOpurpGoseofthistutorialistoshowhowGauianEliminationinLinear AlgebrajrelatestothetheoryofGrobnerbases.LetjK!ֲbGea eld,letm;n>0,andLletLA=(aij )bGeLanm'fn-matrixLwithcoGecientsinK.W*eequiptheringUUP*=K[x1|s;:::;xnq~]withUUthelexicographictermorderingUULexL.a)%W*ritelZaC0oLCoA#programlZRowReduce4?(::: )lZwhichusesrowopGerationsto%bringthematrixAintorowechelonformandthenreturnsthematrix%B=(bij )UUobtainedUUinthisway*.?b)%F*ori=1;:::;m,letfid=ai1Px1/+H 7+HainEʵxn andgid=bi1Px1/+H 7+HbinEʵxnq~.%ShowuthatuGc=fgiDj1im;gi6=0guisuaLex-Grobnerbasisofthe%idealUUI=(f1|s;:::;fm).\Ic)%FindandproveanalgorithmwhichcomputestheLex-Grobnerbasisof%anUUidealUUI7ofPwhichisgeneratedbypGolynomialsofdegreeUU1.?d)%Implement6VyouralgorithminaC0oLCoA%function6VLinearGB06>(::: )6Vwhich%takes'alistofpGolynomialsofdegree'ʲ1generatingI and'returnsthe%Lex-GrobnerUUbasisofUUI.\Ie)%Use3LinearGB/(::: )3to3computetheLex-Grobnerbasesofthefollowing%ideals.)1)8xI1 IJ=8Q(3x1j26x22x3|s;UP2x14x2+4x4|s;UPx12x2x3x4|s)8Q7xQ[x1|s;x2;x3;x4])2)8xI2C=(x1S+8x2+x3|s;UPx1x2|s;UPx1x3|s)Q[x1;x2;x3])3)8xI3C=(x1S+81;UPx2+x3+1;UPx4+x5+1;UPx1+x41)Q[x1|s;:::;x5]x BWō>;1204`2.Gr`obnerTBases-ō>;T utorialT22:ReducedGr@obnerBases궲InOthistutorialweshallimplementanalgorithmto ndthereducedGrobner basis fromanarbitraryone,andweshallstudyvqariousparticularcasesofreducediGrobnerbases.SoletiK bGea eld,n1,P*=K[x1|s;:::;xnq~],r51,궵a?moGduletermorderingon?T^nq~he1|s;:::;ermi,andG=fg1|s;:::;gsF:gPc^rnf0g궲aUU[ٲ-GrobnerUUbasisofthePc-submoGduleM3=hg1|s;:::;gsF:iPc^r1.a)%Implement themethoGddescribedintheproofofTheorem2.4.13.W*rite%a0C0oLCoA!function0ReduceGB./(::: )0which0takesany0[ٲ-GrobnerbasisofM%andUUcomputesthereducedUU[ٲ-Grobnerbasisfromit.?b)%ApplyAyourfunctionAReduceGB-)(::: )Ainthefollowingcases,assumingeach%timeQthatthegivensetsareLex-Grobnerbasesoftheidealstheygenerate.)1)8xG1C=fx^2y+Ky[ٟ^2՗+1;2x^2|syY$+2xy+x;22xyx+y[ٟ^3՗+y[;2y^5K2y^4+7xy[ٟ^2,+8y2gZ=(5)[x;y[ٲ].)2)8xG2C=fxzp^3f\x3y[ٟ^6訸18y[ٟ^412y[ٟ^318y[ٟ^212yl53;k15xy[ٟ^612y[ٟ^57x79y[ٟ^324y[ٟ^267y+zp^3Ÿ26;7y[ٟ^6+6y[ٟ^4+4y[ٟ^3+6y[ٟ^2+4yzp^3Ų+2;7zp^31g7xQ[x;y[;zp].)3)8xG3C=fx^2ֲ+/cy<1;?xy2y[ٟ^2+2y[;?4y^3/c7y^2+/c3y;?1=2x^2ֲ+1=2xy<7xy[ٟ^2,+83=2y1=2gQ[x;y[ٲ].\Ic)%NowڟweequipthepGolynomialringڟP>.withitsstandardgrading(see%Examplep1.7.2).ProvepthatanidealpI,ҸcPishomogeneousifandonly%ifUUitsreducedUU[ٲ-GrobnerbasisconsistsofhomogeneouspGolynomials.%Hint:WFirstshowthatanyhomogeneousidealhasaW[ٲ-Grobnerbasis%consistingUUofhomogeneouspGolynomials.?d)%Letڕm,1,ڕletA=(aij )bGeڕanmn-matrixڕwithcoGecientsinK,%andr?letr?fiK=Hai1Px1ș+L&B+L&ainEʵxn 㽲fori=1;:::;m.r?UsingrowopGerations%only*,gwebringgθAtogreducedrowechelonformgθB3=(bij ),i.e.intherow%echelon\formweclearouteverythingstartingfromthebGottom.F*orthe%non-zeroUBrowsnumbGeredUBi=1;:::;tofBM۲,UBweformthelinearpGolynomials%gid=bi1Px1!=+ʸ;+ʵbinEʵxnq~. IProvethat Ifg1|s=LC#Lex˖(g1);:::;gtV=LC#Lex(gtV)g Iis%theUUreducedLex-GrobnerbasisoftheidealUUI=(f1|s;:::;fm)ofPc.\Ie)%W*rite8aC0oLCoA gprogram8LinRedGB- (::: )8whichcomputesthereducedLex-%Grobner#+ >aisgs>fori=1;:::;t%andgjIJ=bjg13m1'Ͳ+Z[+Zbjgt<mtforjY=1;:::;s.Giveanexampleinwhich%AB0isUUnottheidentityUUmatrix.y RՠWō>;K/2.5Buc9hbAerger'sTAlgorithm9121-ō>;2.5**Buchb`erger'sAlgorithmsрUKnowingN<+andisN;1224`2.Gr`obnerTBases-ō>;궲studied ;inSection2.3.ThenweintroGduceorrecallthefollowingabbrevia- tions.%ӍDe nitionT2.5.1.is~LetBbGethesetB.=f(i;j)j1i L*T9,²(Sij)./Nowweconsidertheelement궵ij =ij pqeyPލ s% k+B=1fk됵"k '2Pc^sɲ.=F*rom=deg;̟qƴ;G"(Pލ ;s% ;k+B=1Dvfk"k)=L*TZ&(Sij )<궲deg#Eqƴ;G0|(ij )wededucethatLF;G(ij )=ij .F*rom(ij )=(ij) dSij =0andUULFt(ij )=ij `MweUUconcludethatUUijisaliftingofUUijinSyzt(GѲ).CorollaryT2.5.3.f(Buc9hbQerger'sTCriterion)|L}'etƵM/|aPc^r beaPc-submodulegeneratedbyƵGa=fg1|s;:::;gsF:gPc^r*nYdf0g,andletG^=(g1|s;:::;gsF:).Thenthefollowingc}'onditionsareequivalent.~oa)%ThesetGisa[-Grobnerb}'asisofM.Hb)%F;orallp}'airs(i;j)2B,wehaveNRpZ;Gʲ(Sij )=0.Pr}'oof.6IfxGisxa[ٲ-GrobnerbasisofMls,thenSij 2MyieldsNRz;G[(Sij )=0byCorollary2.4.9.aandPropGosition2.4.10.a.Conversely*,ifconditionb)󻍑holds,thenSijT΍ G2!|0.UsingPropGosition2.5.2weseethat,foreverypair(i;j)2B,theelementij hasaliftinginSyz.(GѲ).ThusConditionD3|s)ofTheoremUU2.4.1holds.^"궲Letusseehowthiscriterionappliesinpractice.ThefollowingexamplealsomshowsthattheM'le}'adingtermidealofthesquareofanidealis,ingeneral,8- cmcsc10notthesquar}'eoftheleadingtermideal.{ zΠWō>;K/2.5Buc9hbAerger'sTAlgorithm9123-ō>;ExampleT2.5.4.c;1244`2.Gr`obnerTBases-ō>;"궲Itremainstoshowthatwhenthealgorithmstops,thevectorsinthe resultingAtupleAGforma[ٲ-GrobnerbasisofM.DuringtheexecutionoftheproGcedureallpairs(i;j);;2Bareconsidered,sincewhenevers^0i=isincreasedinstep4),allnecessarynewpairs(i;s^09)areaddedtoBq.ByCorollary2.5.3,itsucestoshowthat,forevery(i;j)d2B,wehaveNRt;G (Sij )d=0.IfataFcertainstepFSij =0orNR㹟;G o)(Sij )=0,Fthereisnothingtoprove.FIfataSandg2={xy+=y[ٟ^2L,andletGв=(g1|s;g2).W*ewanttocomputeaGrobnerbasisofMвwithrespGectto"=LexIJandfollowthestepsofUUBuchbGerger'sAlgorithm.1)%LetUUs^0Q=2andBG=f(1;2)g.2)%ChoGoseUU(1;2)2BƲandUUsetBG=;. 3)%W*eUUcomputeUUS12 ?=y[g1S8xg2C=xy^2ꍑ Bg2$ j!Jy^3d=NR;G.(S12x)6=0.4)%Lets^0Q=3,letG^=(g1|s;g2;g3)withg3C=y[ٟ^3L,andletBG=f(1;3);(2;3)g.%ThenUUreturntostep2).2)%ChoGoseUU(1;3)2BƲandUUsetBG=f(2;3)g.3)%W*eUUcomputeUUS13 ?=y[ٟ^3Lg1S8x^2|sg3C=0andUUreturntostep2).} Wō>;K/2.5Buc9hbAerger'sTAlgorithm9125-ō>;2)%ChoGoseUU(2;3)2BƲandUUsetBG=;. 3)%W*e]compute]S23 C="y[ٟ^2Lg2Wxg3 F=y[ٟ^4L.Thenwecalculate]S23ꍑJܴg3$ #@!ꭲ0= %NR4;GA(S23x)UUandUUreturntostep2).2)%SinceUUBG=;,UUwereturntheresultG^=(g1|s;g2;g3)."If:r=1,:i.e.ifMUisanidealinPc,thereisanotheroptimizationofBuchbGerger'sUUAlgorithmwhichturnsouttobeusefulinpractise.PropQositionT2.5.8.qL}'etkG^=(g1|s;:::;gsF:)bekatupleofnon-zeropolynomials,letI=(g1|s;:::;gsF:)Pc,andletti+"=L*Tze(giTL)foriֲ=1;:::;s.Supp}'osethat궲gcd"(tiTL;tj6)=1forsomep}'air(i;j)2B.Thenij hasaliftinginSyz(GѲ).Pr}'oof.6Thisffollowsfromtheobservqationsthatfߵij = "1/&feٟci ;tj6"iC ֱ1"n&feOBcj tiTL"j andzthatUUij = 1K&fe ci*cj:gj6"i, tg1l&fe ci*cjagiTL"jisUUaliftingofij `MinSyzt(GѲ).F8뮍RemarkT2.5.9._uHF*or,sfV;g"2Pc,,sthepair(g[;f)iscalledthetrivialQsyzygy궲of_(fV;g[ٲ)._ThereforePropGosition2.5.8canberephrasedbysayingthatifgcd"(tiTL;tj6)`=1,thenthetrivialsyzygyof(LMjC(gi);LMUw(gj6))canbGeliftedtoUUthetrivialsyzygyofUU(giTL;gj6)."TheUUabGoveresultcanbGeusedtodetectsomespecialGrobnerbases.CorollaryT2.5.10.lL}'et G=fg1|s;:::;gsF:gPvAnf0g, andletI=(g1|s;:::;gsF:).Assumeathatthele}'adingtermsoftheelementsag1|s;:::;gs arepairwiseco-prime.ThenGisa[-Grobnerb}'asisofI.Pr}'oof.6LetG8=9g(g1|s;:::;gsF:).ByPropGosition2.5.8,everyelementij hasaliftingUUinUUSyzt(GѲ).ThusUUGsatis esUUConditionD3|s)ofTheorem2.4.1."궲Finally*,MwecanextendBuchbGerger'sAlgorithminsuchawaythatitnotonlycomputesaGrobnerbasisofasubmoGduleM@)ԵPc^r1,butalsoamatrixofpGolynomialswhichdescribeshowtheGrobnerbasiscanbeexpressedintermsUUoftheoriginalsystemofgeneratorsofUUM.PropQositionT2.5.11.w(TheTExtendedBuc9hbQergerTAlgorithm)UzL}'etjĸG%=LT(g1|s;:::;gsF:)2(Pc^r1)^s bejatupleofnon-zerovectorsinjĵPc^r ;whichgener}'ate1asubmodule1M=hg1|s;:::;gsF:iPc^r1.1WewriteLMxޟt(giTL)=citie iwithciҦ2~ZK9Jn.f0g,ti2~ZT^nq~,and i2~Zf1;:::;rGgfori=1;:::;s.Considerthefollowingse}'quenceofinstructions.~o1)%L}'ets^0Q=s,letAbethes8sidentitymatrix,andletBG=B.~o2)%IfB =;,r}'eturntheresult(Gѵ;A).Otherwise,chooseapair(i;j)2B%anddeleteitfr}'omBq.~o3)%Use)theDivisionAlgorithm1.6.4toc}'omputearepresentation)Sij =%q1|sg1~+ɲ+qs0gs0 +p,wher}'eq1;:::;qs0 2P andp2Pc^r1,suchthat%thec}'onditionsofTheorem1.6.4hold.%Ifp=0,c}'ontinuewithstep2).~ Wō>;1264`2.Gr`obnerTBases-ō>;~o4)%IfОpظ6=0inОstep3),thenincr}'eases^0byОone,app}'endgs0 =صptoG, %addPHf(i;s^09)Zj1i{4_XPލ 8sr0s1% 8k+B=1pqk됲(a1kg1S+8g+8askʵgsF:)`):=4_X(g1|s;:::;gsF:)8(텍Ŵtj33ŭfe)ڟcigcd}(ti*;tja), @ai,hntilʉfe*4Ccjgcd[(ti*;tja).Ӊajoq1a1Sgqs0s17!as0s1)荍):=4_X(g1|s;:::;gsF:)8(a1s0 ;:::;ass0 )tr =a1s0g1S+8g+8ass0 gs궲 nishesUUtheproGof.3"궲T*o showhowthisextendedalgorithmworksinpractice,letusapplyitin theUUsituationofExample2.5.7.ExampleT2.5.12.h`Let)n(Ͳ=2,)letro=1,)letM?P\=K[x;y[ٲ]bGe)theidealgeneratedby싵g1?=x^2 handg2=xy+y[ٟ^2L,andlet싸GZ=(g1|s;g2).AsinExample2.5.7,wefollowthestepsoftheBuchbGergerAlgorithm,exceptthatweUUnowusetheextendedversionabGove.1)%LetUUs^0Q=2,UUletA=b \o1F0⾍\o0F1 b1c,UUandletUUBG=f(1;2)g.2)%ChoGoseUU(1;2)2BƲandUUsetBG=;.3)%W*ecomputeS12 ?=xy[ٟ^2d=0<g1[+(y[ٲ)g2+y[ٟ^3βandletq1C=0,q2=y[ٲ,%andUUp=y[ٟ^3L.4)%Lets^0Q=3,letG^=(g1|s;g2;g3)withg3C=y[ٟ^3L,andletBG=f(1;3);(2;3)g.%W*eSappGendthecolumnvectorSy[a16xa20a1+y[a2etoSthematrixSA%andUUgetUUA=b h1F0 Fy⾍\o0F1x+y)b..Thenwereturntostep2). O<2)%ChoGoseUU(1;3)2BƲandUUsetBG=f(2;3)g.3)%W*eUUcomputeUUS13 ?=y[ٟ^3Lg1S8x^2|sg3C=0andUUreturntostep2).2)%ChoGoseUU(2;3)2BƲandUUsetBG=;.3)%W*eS%computeS%S23 ?=y[ٟ^4d=04g1+0g2+y[g3|s.S%Thenwereturntostep2).2)%SinceBȵBF=Rո;,Bwereturntheresult(Gѵ;A),whereGꦲ=R(g1|s;g2;g3)Band%A=b h1F0 Fy⾍\o0F1x+y)b.. ϟWō>;K/2.5Buc9hbAerger'sTAlgorithm9127-ō>;%Exercise1.RLetP1)=FK[x;yR;zc],letGo==F(x-=28yR;xyzc)F2PH-=2s,and %let}=@4DegRevLex.z.}P9erformallstepsofBuchbAerger'sAlgorithmapplied%to~G.~Then ndatermorderingEsuc9hthatGisaR-Gr`obnerbasisof%theTidealT(x-=288yR;xy`zc).%Exercise2..ApplyBuc9hbAerger'sAlgorithmasinExample2.5.7tocom-%putedaDegLexPos-Gr`obnerbasisofthesubmoAduledM=hg1*;g2;g3;g4i%ofTQ[x;yR]-=3?inTthefollo9wingcases. *?a)8xg1m=(x-=2*;xyR;y-=2}Q),Tg2m=(y;0;x),Tg3m=(0;x;y),Tg4m=(y;1;0))b)8xg1m=(y`8x;yR;y),Tg2m=(xy;x;x),Tg3m=(x;y;y),Tg4m=(x;y;0)*6c)8xg1m=(0;yR;x),Tg2=(0;x;xy`8x),Tg3=(yR;x;0),Tg4=(yR-=2}Q;yR;0)%Exercise+3.KInZthecasesofExercise2,determinerepresen9tativesZfora%K-basisTofTQ[x;yR]-=3*=|rM.%Exercise 4.yFindtoutwhic9hmoAduletMQ[x;yR]-=3finExercise2contains%theTv9ector&rm=(x=2*y`8yR=2+xyR=2}Q;TxyR=2yR=2+x=28+2xy`xyR;Tx=2*y+xyR=23xy+x)%Exercise5.AIbpAolynomialIpfJ2P1=K[x1*;::: ;xn7]isIpcalledabinomial %ifitisoftheformfq=at+a-=0t-=0dwitha;a-=0C2KLnf0gandt;t-=0C2T-=n7.Let%bAe.atermorderingon.T-=neandIWa.binomial ^ideal,i.e.anidealgenerated%b9yTbinomials. *?a)7xPro9veTthatthereducedTR-Gr`obnerbasisofIɯconsistsofbinomials.)b)7xGiv9eniatermit2T-=n7,ishowthatiNF᠟ӎ;I(t)iisascalarmultipleofa7xterm.%Exercisen<6. "Consider^;thepAolynomialring^;P=Q[x;yR],thePH-sub-%moAdule'M=hg1*;g2;g3;g4iPH-=3 _suc9h'that'g1D=(xyR;x;y),'g2D=(y-=2+y;%x +yR-=2}Q;x),g3=W(x;yR;x),g4=W(yR-=2}Q;yR;x),andthemoAduletermordering%p=LexPosJ. *?a)7xUsing2thealgorithmgiv9eninPropAosition2.5.11,computea2R-Gr`obner7xbasis"fg1*;::: ;gfhsG0:gofM,"wheres-=0e:s4,andamatrixAsuc9hthat7x(g1*;::: ;gfhsG0:)=(g1;::: ;g4)8A.)b)7xNo9wusethemethoAddescribedintheproofofProposition2.4.13to7xcomputeLthereducedLR-Gr`obnerbasisfg-=R0h1*;::: ;g-=R0h6gofM.LThen nd7xaTmatrixTA-=0suc9hthat(g-=R0h1*;::: ;g-=R0h6)=(g1*;::: ;g4)8A-=0.*6c)7xF:orthefollo9wingelementsofPH-=3s,checkwhethertheylieinM,andif7xtheyZ]do, ndtheirrepresen9tationsintermsofbAothZ]fg-=R0h1*;::: ;g-=R0h6gand7xfg1*;::: ;g4g.<1)J?qm1m=(2yR;y`81;xy+8yR)<2)J?qm2m=(xyR-=58xy`+yR;xy-=4+8x+2yR-=2yR;y-=5+8xy))d)7xF:or.Sthefollo9wingpairsofelementsof.SPH-=ruS,checkwhether.Sm13+ɌM7xagreesTwithTm28+8M8intheresidueclassmoAdulePH-=ruS=|rM.<1)J?qm1m=(2yR;x-=2*y +Tx-=2+xy+2x3yR;x+y),bm2m=(x-=2+y x;I?qx-=38+82x-=2*;x-=2yR)<2)J?qm1m=(x-=3+Xx-=2+y)x;x-=2+x;x+yR),ydm2m=(y;x-=3+2x-=2xy)yR;0) 湠Wō>;1284`2.Gr`obnerTBases-ō>;T utorialT23:Buc9hbQerger'sTCriterion궲InthistutorialweshallimplementBuchbGerger'sCriterion2.5.3anduseit to3decidewhethercertainsetsofpGolynomialsareGrobnerbasesoftheidealstheygenerate.Asinthewholesection,weletõKK߲bGea eld,weletõP?=궵K[x1|s;:::;xnq~]ɯbGeɯapolynomialringoverɯK,ɯweletɯ%bGeamoduletermorderingonT^nq~he1|s;:::;ermi,whererv\/?1,weletG=/?(g1|s;:::;gsF:)bGeatupleofjnon-zerovectors,andweletjMD,Pc^r cbGethePc-submoGdulegeneratedbytheUUvectorsinUUGѲ.a)%W*ritePaC0oLCoA$functionPCheckGB*;(::: )PwhichtakesPGv!andusesBuch-%bGerger'sCriterion2.5.3tocheckwhetheritformsa[ٲ-Grobnerbasis%ofM.(Hint:Y*oumaywanttousethefunctionNormalRemainderSjϲ(::: )%fromUUT*utorial15orthebuilt-inC0oLCoA!ϡfunctionUUNR(::: ).)?b)%LetUG=fx29zx^2l1|s;x3x^3l1|sgQ[x1;x2;x3g.UUsethefunctionUCheckGB((::: )%tocheckwhetherGisa[ٲ-Grobnerbasisoftheidealitgenerates,where%薲isoneofthefollowingtermorderings:Lex,DegLex,OrdL(V8)where@%V=`q &b0; &b0܍ &b1qI0;I1܍I0q 01; 00܍ 00&q `0BorUUV=`q &b0; &b0܍ &b1qI1;I0܍I0q 00; 01܍ 00&q `-i.\Ic)%Use5thefunction5CheckGB)e (::: )5todeterminewhichofthefollowingsys-%temsqofgeneratorsareGrobnerbaseswithrespGecttothestatedtermor-%deringsNgoftheidealsandmoGdulestheygenerate.Inthe rstthreecases,%tryto ndatermorderingandasystemofgeneratorscontainingGsuch%thatUUCorollary2.5.10canbGeapplied.)1)8xGrS=fx1|sx^2l2Ѹ}^x1x3+}^x2;x1x2}^x^2l3;x1}^x2x^4l3grSQ[x1|s;x2;x3]with7xrespGectUUtoUULex)2)8xGF=fx^4l1|sx^2l2pkx^5l3;x^3l1x^3l2k1;x^2l1x^4l2k2x3gFQ[x1|s;x2;x3]withrespGect7xtoUUDegLex)3)8xGV=fx1|sx3slx^2l2;x1x4x2x3;x2x4x^2l3gVQ[x1|s;x2;x3;x4]r|with7xrespGectUUtoUUDegRevLex)4)8xG=f(x^2l1lnx2|sx3)(e1+ne2);(x1x3nx2x4)(e1ne2);(x^2l3nx1x4)e1;7x(x^2l3tƸSx1|sx4)e2gQ[x1|s;x2;x3;x4]^2 withtrespGecttotPosDegRevLex7xandUUDegRevLexPos)5)8xG=f(x1-Xx^2l2|s)e1;(x1Xx^3l3)e1;(x2Xx^2l1)e2;(x2Xx^3l3)e2;(x3Xx^2l1)e3;7x(x32@x^3l2|s)e3g6Q[x1|s;x2;x3]^3 GڲwithgrespGecttogPosDegRevLexJand7xDegRevLexPos?ֲd)%Let~n>1,~andletWbGethelexicographictermorderingonK[x1|s;:::;xnq~;%y1|s;:::;ynq~]suchthatx1A> 'и> 'еxn 6>y1A>>ynq~.Moreover,%fordi=1;:::;n,dletsid=P US1j1 < Fxj1 `6xji {bGedthei^th elementary%symmetricHpGolynomialinHx1|s;:::;xn AƲ(seealsoT*utorial12),andlethi;j 2=%P0?ڟ ja+ + nl=iaxW j4=j o:x^ n፴nRxfor*i;j[=k1;:::;n.*UseBuchbGerger's*Criterion%toUUprovethatthepGolynomials`:gid=(1)iTL(yi,8si)+ei1 X tmjg=1(1)j6hijT;i(yjosj) AWō>;K/2.5Buc9hbAerger'sTAlgorithm9129-ō>;%such that iG=1;:::;nform a[ٲ-GrobnerbasisofthepGolynomialideal %I=(y1S8s1|s;:::;yn^snq~).\Ie)%V*erifyutheresultofd)forun=1;:::;5byuapplyingyourfunction%CheckGBJq(::: ).UUCanyoucomputethisforlargerUUn?Howfarcanyougo?+{T utorialT24:ComputingSomeGr@obnerBases궲The̜purpGoseofthistutorialistoimplementa rstversionofBuchbGerger'sAlgorithminthecaseofpGolynomialideals,andtouseittostudysomeparticularlexamples.F*orinstance,wewillseethattheelementsofthereducedGrobnerbasisofanidealcanhaveveryhighdegree,evenifthegeneratorsofUUtheidealshaveUUlowdegrees."Then,forthespGeci cidealꈵI=(y[z Wzp^2 ;xzzp^2 ;xy0zp^2),youwillbGeguided to ndallpGossiblereducedGrobnerbasesof I,andtogiveameaningtoathepictureonthecoveraofthisbGook.aAsusual,weletaյP?|=K[x1|s;:::;xnq~]bGeUUapolynomialringoverUUa eldUUK.a)%W*riteRaC0oLCoA!functionRSPoly8C(::: )Rwhichtakesatupleofnon-zeropGoly-%nomials|w(g1|s;:::;gsF:)and|windicesi;jY2f1;:::;sgwithi6=jas|warguments%andoreturnstheS-pGolynomialoSij zofgiandgjJwithorespecttothecurrent%termUUordering.?b)%Implement BuchbGerger'sAlgorithm2.5.5inthecaseofpolynomialideals.%T*othisend,writeaC0oLCoA#Z#programFirstGB)ځ(::: )whichtakesatuple%ofU-non-zeropGolynomialsgeneratingtheidealandcomputesaGrobner%basiswithrespGecttothecurrenttermordering.(Hint:F*orstep3),use%theUUbuilt-infunctionUUNR(::: )orNormalRemainderS((:::)ofUUT*utorial15.)\Ic)%UsingFirstGB)a(::: ),calculatetheGrobnerbasesofthefollowingideals%withUUrespGecttothestatedtermorderings.)1)8xI=(x^2l2|s;UPx1x2x3S+8x^3l3)Q[x1;x2;x3]UUwithUUrespGecttoDegRevLex)2)8xI=>(x^2l1|sx25F1;UPx1x^2l25Fx1)>Q[x1|s;x2]withrespGecttoLex_ײand7xDegLex)3)8xI=N(x1W;x^4l3|s;UPx2x^5l3|s)NQ[x1;x2;x3]withrespGecttoLexsand7xDegRevLex?ֲd)%ProveUUthatforeverynumbGerUUm1,UUthereducedGrobnerbasisof%Im _=(x䍴m+11.uwx2|sx䍴m13x4;UPx1x䍴m12Jxm፱3;UPxm፱1x3xm፱2x4|s)K[x1;x2;x3;x4]&%with'respGectto'DegRevLex63contains'fm _=x䍴mr2 +13E x^mr2l2 x4|s.Notethatthe%degree+om^2a+1of+othispGolynomialismuch+ohigherthanthedegreesofthe%generatorsLXofLXIm.CanyouwritedownthewholereducedGrobnerbasis%ofUUIm withUUrespGecttoDegRevLex4:?(GuessitorproveUUit!)\Ie)%Ifyoucouldn'tdothesecondpartofd),calculatethereducedGrobner%basisCoftheidealCIm ?withrespGecttoDegRevLex7-usingFirstGB)(::: )for%m=1;:::;100UUandUUdetermineitslength. ؠWō>;1304`2.Gr`obnerTBases-ō>;f)%ProvethattheidealI3 $ofpartd)hasthesamereducedGrobnerbases %withUUrespGecttoUULexjandUUDegRevLex4:.DoesthisholdforallUUm1?"Intheremainderofthistutorial,wewanttostudythepGolynomialideal궵I=(xylϸzp^2 ;xzzp^2;y[zzp^2)A`inP*=K[x;y[;zp].A`Althoughwearenotgoingtouseit,wementionthatߵImistheidealofallpGolynomialswhichvqanishatthreelinesinA^3bK passingthroughtheorigin,or,equivqalently*,atthreepGointsinUUP^2bK (seeUUT*utorials27and35).g)%Let2ѲbGeanytermorderingsuchthatx'> zGandy>zp.Showthat%theUUreducedUU[ٲ-GrobnerbasisofI7isfxzw8zp^2 ;y[zzp^2 ;xyzp^2g.?h)%Let2ѲbGeanytermorderingsuchthatx'> zGandz>y[ٲ.Showthat%theUUreducedUU[ٲ-GrobnerbasisofI7isfxy8y[zp;xzwyzp;z^2%8yzg.i)%Let2ѲbGeanytermorderingsuchthaty> zGandz>x.Showthat%theUUreducedUU[ٲ-GrobnerbasisofI7isfzp^2%8xzp;y[zwxz;xyxzg.j)%Considerthesituationwheresrisatermorderingsuchthatz7>)xand%z'> x&y[ٲ.ShowthatthereareonlytwopGossiblereducedGrobnerbases%of I, accordingasx> bylory[>x. 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"r.盟.5.P..]7.:..q.꟥m9.c .ܟ.UK;.kΟ.IG.&)=.9ȓ.Ჟg.+?..zE.WA.5.#.C.zb..lE.e埞@.C^. ןG.P.ɟ.B]I..t4.Q;K./&ڡ. y.M.柚.柚.p1?xj.9y埂6z )(1;]0;0)l I1(0;]1;0)x(0;]0;1) &PWō>;K/2.5Buc9hbAerger'sTAlgorithm9131-ō>;T utorialT25:SomeOptimizationsofBuc9hbQerger'sTAlgorithm궲The purpGoseofthistutorialisto ndandtoimplementoptimizedversions ofBuchbGerger'sAlgorithminthecaseofpolynomialideals.TheamountoftimeconsumedbyacertainGrobnerbasiscomputationdepGendslargelyon thenumbGer ofpairswhichhavetobGedealtwith,andonthenumbGerofreductionstepswhichhavetobGeperformedinordertotreateachpair.Thereforegwewillaskyoutoimplementc}'ounters: inyourprogramswhichmeasure/thesequantities,andwewilljudgeourprogresstowardsourgoalofoptimizingBuchbGerger'sAlgorithmbyloGokingatthenumbGersreturnedbythoseUUcounters."Let˵P*=K[x1|s;:::;xnq~]bGeapolynomialringovera eld˵K,letIP`ZbGean1ideal,andlet1GW=O(g1|s;:::;gsF:)bGe1atupleofnon-zeropolynomialswhichgenerateI.F*urthermore,let bGeatermordering,andlettheelements궵tiTL;tij 2T^nq~,ij2Pc^sɲ,andSij2PXzbGede nedasatthebeginningofthissection.a)%UpGdate=yourC0oLCoA!function=FirstGB(߲(::: )=fromT*utorial24suchthatit%returnsnotonlya쮵[ٲ-GrobnerbasisofI,butalsothenumbGerofpairs%(i;j)suchthatSij v6=k0,i.e.suchthatthenormalremainderhadtobGe%computed,2andthetotalnumbGer2ofreductionstepswhichwerenecessary%toUUcomputeallthosenormalremainders.%Hint:iԲY*ouwillhaveitomoGdifythefunctioniNormalRemainderS)(::: )ifrom%T*utorialUU15suitably.?b)%ApplyyournewfunctionFirstGB)M(::: )inthefollowing vecases.Each%time, computeaGrobnerbasiswithrespGectto DegRevLex7uandonewith%respGectUUtoUULexL.)1)8xI=(x^2l1S82x^2l2+3x1|s;UPx^3l12x1|sx2)UUinQ[x1|s;x2])2)8xI=(x1S82x^4l3|s;UPx23x^5l3|s)UUinQ[x1;x2;x3])3)8xI=(x^2l1S82x^2l2|s;UPx^3l13x^3l3|s;UPx^4l1x^4l4|s)UUinQ[x1;x2;x3;x4])4)8xI=(x^3l1S84x^3l2|s;UPx^5l17x^5l3|s;UPx^7l111x^7l4|s)UUinQ[x1;x2;x3;x4])5)8xI=(x^2l1S+8x^3l2+x^3l31;UPx^3l1+x^4l2+x^5l31)UUinQ[x1|s;x2;x3]\Ic)%Implement(aC0oLCoA"Gfunction(SecondGB.(::: )(whichtakesthelist(GXand%computesBaB[ٲ-GrobnerbasisofI ˲viaBuchbGerger'sBAlgorithm2.5.5,%wherezthepairz(i;j)GS2BUiszchoseninstep2)accordingtothenor-%mal]zselectionstrategy(seeRemark2.5.6.c),andwheretheoptimization%whichUUfollowsfromPropGosition2.5.8isused.?d)%Apply| yourfunction| SecondGB-{(::: )| inthecasesofb)andcomparethere-%sults5ofyourcounterswiththosereturnedbythefunction5FirstGB( (::: ).\Ie)%Given1-i;1324`2.Gr`obnerTBases-ō>;f)%ProveSthatonecandropapairS(i;j)inStheexecutionofBuchbGerger's %AlgorithmifitiscontainedinaBuchbGergertripleandiftheothertwo%pairs[havebGeentreatedalready*.WriteaC0oLCoA#yfunction[ThirdGB*(::: )%whichisbasedonSecondGB/(::: )andaddsthisnewoptimization.T*o%make$surethatyoudonotdropmorethanonepairfromaBuchbGerger%triple,ȷimplementalistȷT,FwhichkeepstrackofthepairswhichhavebGeen%treatedUUalready*.g)%ApplyyourfunctionThirdGB)d(::: )inthecasesofb)anddeterminethe%improvementUUwhichhasbGeenachieved.?h)%Start'againwithyourimplementation'SecondGB/(::: )'ofBuchbGerger's%Algorithm,andreplacestep4)bythefollowingsequenceofinstructions.$4a)7xIncrease1s^0òby1one.AppGendgid=NR;G.(Sij )toGѲ,andformtheset7xC~4=f(i;s^09)j1i;)2.6HilbAert'sTNullstellensatz9133-ō>;2.6**Hilb`ert'sNullstellensatz胅TheN;1344`2.Gr`obnerTBases-ō>;"궲Givena eldextensionK SȵL,animpGortantresultisprovedabGout thenbGehaviourofidealsunderextensionfromnK[x1|s;:::;xnq~]toL[x1;:::;xnq~].ThisresultisthekeytotheweakformofHilbGert'sNullstellensatz(seeTheo-remb2.6.13)whichprovidesuswithane ectivewaytocheckwhetheragivenpGolynomialAidealhaszerosinA}fe 5UKo7n൲,whereA}fe 5UKײisthealgebraicclosureofAK.F*or9^ourideal9^(x^4}e+2x^2+1)R[x],9^theeasyobservqation9^1㊵=2 8ܲ(x^4}e+2x^2+1)sucesUUtoconcludethatithaszerosinthealgebraicclosureUUCofR."Finally*,qDweprovetheNullstellensatzinitsfullgenerality(seeTheo-rem8}2.6.16).ItsaysthattheopGerationofformingthevqanishingidealofasubsetof}fe 5UKnisaninversetotheopGerationoftakingthesetofzerosofapGolynomialidealifoneconsidersradicalidealsonly*.Thishighlightstheim-pGortanceloftheidealtheoreticoperationofformingtheradicalofanideal.InGGthecaseoftheprincipalidealGG(x^4V+,2x^2+1)GGinC[x],GGitsaysthatthevqanishingUUidealofitssetofzerosUUfi;igisUUitsradicalidealUU(x^2S+81).2.6.A TheTField-TheoreticV ersion궲LetEKtabGeEanarbitrary eld.Manyalgebraicgeometersusethefollowingterminology*.De nitionT2.6.1.is~Ah@ nitelyhgeneratedhK-algebraisalsocalledanane궵K-algebra."AccordingvtoCorollary1.1.14,suchalgebrasareoftheformvPV=IsXforsomepGolynomial0ring0Pi=4ڵK[x1|s;:::;xnq~]andsomeidealI4ڵPc.Nowwepresentthree`lemmasleadinguptothe rsttheoremofthissectionwhichisalsocalledHthe eld-theoreticversionofHilbGert'sNullstellensatz.Recallthatthe eldoffractionsofapGolynomialringƵP=4)K[x1|s;:::;xnq~]isusuallydenotedbyUUQ(Pc)=K(x1|s;:::;xnq~).LemmaT2.6.2.\L}'etxbeanindeterminateoverour eldK.ThenK(x)isnotananeK-algebr}'a.Pr}'oof.6SuppGose[[K(x)"=K[3sf133);fe۟g1 4A;:::;fs۟);fegs ђ][[for[[somef1|s;:::;fsF:;g1|s;:::;gs\2"K[x]Bsuchthatg1.#g2'|jgs R6=0.Since Ia1(&fexɵ= W2K[x],wemayassumeg1.#g2'|jgs)ĵ= R2K.Then,thefraction D1_&fe*=1+g1 g2  gs7(canbGewrittenasapolynomialexpressionin!)f1);fe۟g1;:::;fs۟);fegs ђ.Clearingdenominators,weget(g1()g2'|jgsF:)^id=(1+g1g2'|jgsF:)h O<궲forQsuitableQi>0andh2K[x].QNowK[x]isQafactorialdomain(seeTheo-rem(1.2.13),butclearlynoirreduciblefactorofthenon-constantpGolynomial1+g18Bg2'|jgs]candivideoneofthepGolynomials̵g1|s;:::;gsF:.Thiscontradic-tionUU nishestheproGof.ᥝLemmaT2.6.3.\L}'etABGCKbethreerings.~oa)%IfhgBishga nitelygener}'atedA-module,hgthenitisalsoa nitelygener}'ated%A-algebr}'a. ?Wō>;)2.6HilbAert'sTNullstellensatz9135-ō>;Hb)%IfmBisma nitelygener}'atedA-algebramandifC$isa nitelygener}'ated %Bq-algebr}'a,thenCKisa nitelygeneratedA-algebra.TPr}'oof.6LetKfb1|s;:::;bsF:gbGeKasetofgeneratorsofKBasanA-moGdule.Then궵Bϲ=\^Ab1Wo+`+AbsA[b1|s;:::;bsF:]BimpliesHclaima).F*ortheproGofof/b),weuseCorollary1.1.14towrite/B=3A[x1|s;:::;xnq~]=Iܲwithanideal궵IA[x1|s;:::;xnq~]andC~4=Bq[y1|s;:::;ym]=J withanidealҵJQBq[y1|s;:::;ym].ThenUUtheclaimfollowsfromT#̵CT͍~4+3~4= jA[x1|s;:::;xnq~;y1;:::;ym]=bWI¸8A[x1;:::;xnq~;y1;:::;ym]8+[ٟ1 M(J9)b궲wherep޵Pղ:A[x1|s;:::;xnq~;y1;:::;ym] r!Bq[y1;:::;ym]pispthecanonicalhomo- morphism."궲TheZnextresultisdeepGer.W*ewantZtoshowthatundercertaincircum-stances,a,K-subalgebraofananeK-algebraisananeK-algebra.ThefollowingUUexampleshowsthatthisisnotalwaysthecase.ExampleT2.6.4.c;1364`2.Gr`obnerTBases-ō>;궲everyxproGductx iTL jcbyanelementofxA0|s 1+PvK+PvA0 tV.Ifweiteratethose substitutions,QweseethateveryelementofQBisinA0|s 1+1: +1:A0 t+A0|s,i.e.UUtheUUA0|s-moGduleBƲisgeneratedbyf1; 1|s;:::; tVg.YzTheoremT2.6.6.dH(Field-TheoreticTV ersionofHilbQert'sNullstellen-satz)~L}'etP*=K[x1|s;:::;xnq~]andmamaximalidealofPc.~oa)%F;oreveryi2f1;:::;ng,theinterse}'ctionm6\K[xiTL]isanon-zer}'oideal.Hb)%TheaneK-algebr}'aPV=misa nitelygener}'atedK-vectorspace.Pr}'oof.6First$weshowthata)impliesb).Byassumption,for$ip=1;:::;n,theintersectionm\K[xiTL]isanon-zeroprincipalidealgeneratedbysomenon-constant pGolynomial fiKe2K[xiTL].Thentheidealn=(f1|s;:::;fnq~)P궲is!containedin!mand!wehaveasurjectivehomomorphism!PV=nk kP=m.ThereforeqitsucestoshowthatqPV=nisqa nitelygeneratedqK-vectorspace.Nowhff1|s;:::;fnq~gishaGrobnerbasisofhnwithhrespGecttoeverytermordering궵 byGCorollary2.5.10.IfwewriteGL*T)֟l(fiTL)=x:di;Zi ZwithGdid1fori=1;:::;n,weMmaydeducefromCorollary2.4.11thataMK-vectorspacebasisofMPV=nisgivenbythe nitesetofresidueclassesofthetermsx: 1l1 Dx^ n፴nݲsuchthat0 id1,wedenotetheaneеK-algebraPV=mbyBq.Leti2f1;:::;ng,letП~fegx ibGetheresidue[classof[xi3inBq,[andlet[AbGe[the eldoffractionsoftheintegraldomainTK[~fegxi h]containedTinthe eldTBq.Clearly*,consideredasanA-alge-bra,WB%isWgeneratedbyf~fegx1 3;:::;~fegxaği@1;~fegxaği+#1;:::;~fegxağn Bg.WByinductionandtheimplicationq-shownabGove,q-Bisa nitelygeneratedA-vectorspace.Therefore,byLemma2.6.5,the eldAisa nitelygeneratedK-algebra.Then~fegx UiHisnotanukindeterminateoverukK,byukLemma2.6.2,andhencemND\K[xiTL]isukdi erentfromUU(0)."궲ConditionCa)oftheabGoveCtheoremdoesnotholdformoregeneralrings,asthe1followingexampleshows.(InthisexampleweshallusesomefactsabGoutpGowerseriesandLaurentseriesringswhichwillbGediscussedmorethoroughlyinUUChapterV.TheinexpGeriencedreadermaysafelyskipit.)ExampleT2.6.7.c;)2.6HilbAert'sTNullstellensatz9137-ō>;궲wherec;a1|s;:::;as 2õKAٲand 1|s;:::; s 2Naresuchthata1|s;:::;as are pairwiseUUdistinctandUU 1S+8g+8 s R=d."F*orl0instance,theF undamen9talTheoremofAlgebral0saysthatthe eldofcomplexnumbGersCisalgebraicallyclosed.The eldsRandQarenot&algebraicallyclosed,sincethequadraticpGolynomial&ڵx^2X^+1is&irreducibleoverUUthem."AnYimpGortantresultwhichyoushouldknowisthat,forevery eldYK,thereexistsanalgebraicextension eld蕟}fe 5UKwhichisanalgebraicallyclosed eld.(Andifyoudonotknowit,youcanloGokforinstanceat[La70@],Ch.7.)The% eld%}fe 5UKisuniqueuptoa%K-algebraisomorphismandiscalledthealge-braicclosure#ofK.F*orexample,the eldCisthealgebraicclosureofR,sinceitisalgebraicallyclosedandanalgebraicextensionofR.ThealgebraicclosureUUofUUQisthe eldofalgebraicnumbGersUUCfeQqȲdiscussedinT*utorial18."The eld-theoreticversionofHilbGert'sNullstellensatzcanalsobeinter-preted0asastructuretheoremformaximalidealsinpGolynomialringsoveralgebraicallyUUclosed elds.CorollaryT2.6.9.fL}'et0K0~beanalgebraicallyclosed eld,andlet0mb}'eamax-imal^Zide}'alin^ZK[x1|s;:::;xnq~].Thenthereexistelements^Za1|s;:::;an inKvsuchthat[m=(x1S8a1|s;:::;xn^anq~)Pr}'oof.6TheoremB2.6.6yieldsnon-zeropGolynomialsBf1|s;:::;fn82msuchBthat궵fiܸ2dK[xiTL]fori=1;:::;n.EverypGolynomialѵfifactorizescompletelyintolineargfactors,sincegKisalgebraicallyclosed.Moreover,gtheidealmismaxi-mal,Yhenceprime.ThisimpliesthatitcontainsoneofthelinearfactorsofeachpGolynomialqfiTL,qsayxieai.qThenmcontainstheideal(x1a1|s;:::;xnanq~)which,wontheotherhand,isamaximalideal.ThustheymustbGeequalandtheUUproGofiscomplete. -%2.6.B TheTGeometricV ersion궲IntheremainderofthissectionwewanttoexplainthegeometricversionsofHilbGert'sLNullstellensatz.TheGermanword\Nullstellensatz"literallymeans\zero-places-propGosition".UULetusde newhatthisrefersto.De nitionT2.6.10.o3|LetL׵K~4LbGeLa eldextension,letLן}fe 5UKbethealgebraicclosureUUofUUK,andletP*=K[x1|s;:::;xnq~].a)%Anyelementy(a1|s;:::;anq~)2L^n \(whichyweshallalsocallapQoin9tofyL^nq~)is%saidGtobGeazeroofapolynomialGfڧ2P0inL^n iff(a1|s;:::;anq~)=0,Gi.e.%iftheevqaluationof׵ffatthepGoint(a1|s;:::;anq~)iszero.Thesetofallzeros%of۵f-jinL^n YwillbGedenotedby۸ZLt(f).Ifwesimplysaythat(a1|s;:::;anq~)%isUUazeroofUUf,wemean(a1|s;:::;anq~)2}fe 5UK Zn andf(a1|s;:::;anq~)=0. Wō>;1384`2.Gr`obnerTBases-ō>;?ֲb)%F*orUUanidealUUIPc,thesetTofzerosUUofIsinL^n Ӳisde nedas?{ZLt(I)=f(a1|s;:::;anq~)2Ln8jf(a1;:::;anq~)=0UUforall fڧ2Ig%AgainwecallthesetofzerosofݵIin}fe 5UK{ӟn.simplythesetofzerosofݵI %and7:denoteitby7:Z_(I).Laterweshallalsocall7:Z_(I)the7:anev\rariet9y%de nedUUbyUUI."ItťiseasytoseethatthesetofzerosťZLt(f)ofťapGolynomialťf@2,P궲inヵL^n UagreeswiththesetofzerosZLt((f))oftheprincipalidealitgenerates.Moreover,ifanidealIPisgeneratedbyasetofpGolynomialsff1|s;:::;fsF:g,thenUUwehaveUUZLt(I)=\^s;Zi=1 tOZL(fiTL)."Algebraically*,PthesetofzerosofanidealcorrespGondstoasetofmaximalidealsUUinthepGolynomialring,asournextpropositionshows.PropQositionT2.6.11.wL}'etyK|beyanalgebr}'aicallyyclosed eld,letyI[beaproperide}'alinP*=K[x1|s;:::;xnq~],andlet<(b}'ethesetofmaximalidealsinP *whichc}'ontainI.Thenthemaph<':Z_(I)!*de ne}'dby'(a1|s;:::;anq~)=(x1S8a1;:::;xn^anq~)isbije}'ctive.Pr}'oof.6F*or?p=(a1|s;:::;anq~)2K^n(,?wedenoteby?mpfj=(x1*a1|s;:::;xn5anq~)the correspGondingmaximalidealin Pc.Thenthemap'canbGedescribedby궵'(p))=mpR.Firstweprovethat'iswell-de ned.F*orapGointp)2Z_(I),allpGolynomials in Ivqanishatp.UsingtheDivisionAlgorithm1.6.4,wethenseeUUthatthosepGolynomialsbelongtoUUmpR."The)map)'isclearlyinjective.Henceitsucestoshowthatitissurjec-tive.W*echoGoseamaximalidealm^2.ByCorollary2.6.9,thereexistsapGointp=(a1|s;:::;anq~)2K^n Fsuchthatm=mpR.Bythede nitionof,wehaveUUmpfjI.UUItfollowsthatp2Z_(I),UUandtheproGofiscomplete.#K*"궲OurnextresultisusefulforcomparingthesetofzerosofIҲinL^n nfordi erentCextension eldsCLofK.CRemembGerthatifCK~4LisCa eldextensionand۵IȽisanidealofP*=K[x1|s;:::;xnq~],weusethenotationIL[x1|s;:::xnq~]todenoteUUtheidealofUUL[x1|s;:::xnq~]generatedUUbythesetUUI.PropQositionT2.6.12.wL}'etKm/QLbea eldextensionandIanidealof궵K[x1|s;:::;xnq~].ThenotIL[x1|s;:::;xnq~]8\K[x1;:::;xnq~]=IIndp}'articular,wehavedIL[x1|s;:::;xnq~]=L[x1;:::;xnq~]difdandonlyifwehave 궵I=K[x1|s;:::;xnq~].Pr}'oof.6Obviously.weonlyneedtoprovethattheleft-handsideiscontainedinVI.VW*echoGoseatermordering/onT^n 3ԲandletG'm=fg1|s;:::;gsF:gbGea~F[ٲ-Grobner~FbasisofI.F*romLemma2.4.16itfollowsthatthesetGis iWō>;)2.6HilbAert'sTNullstellensatz9139-ō>;궲alsosas͵[ٲ-GrobnerbasisoftheidealIL[x1|s;:::;xnq~].Nowletf\bGeapoly- nomialinIL[x1|s;:::;xnq~]'I\K[x1;:::;xnq~].IfwecomputethenormalformNF!՟'Tk(f)^Yusing^YtheDivisionAlgorithm1.6.4,weonlypGerformoperationsinside˵K[x1|s;:::;xnq~],andthereforefAg-زNF4(f)isintheidealgeneratedin5K[x1|s;:::;xnq~]by5thesetofpGolynomials5fg1;:::;gsF:g,whichis5I.But궵fڧ2IL[x1|s;:::;xnq~]UUimpliesNF\t (f)=0,UUhencewehaveUUfڧ2I.551Z"궲The-4questionswhichidealshavezerosandhowonecancheckthatarenowUUansweredbythefollowingtheoremanditscorollary*.qTheoremT2.6.13.jUF(W eakTNullstellensatz):L}'eẗ́Kbë́a eld,andletIfb}'eaproperidealof̈́P=/wK[x1|s;:::;xnq~],i.e.let궵IPc.ThenZ_(I)6=;.Pr}'oof.6Let^}fe 5UKbGe^thealgebraicclosureof^K t,andlet}feP=}fe 5UK [x1|s;:::;xnq~].Then6I}feP s5is6apropGeridealof}fePbyPropGosition2.6.12.Sinceweknowthat6}feP궲is.NoGetherian,theideal.I}feP -iscontainedinamaximalidealmof}fePybyPropGo-sitionA 2.4.5.c.NowCorollary2.6.9saysthatthereisapGointA (a1|s;:::;anq~)2}fe 5UK Zn궲such that m=(x1$a1|s;:::;xn$anq~).Hence (a1|s;:::;an)is azeroofm,andthereforeUUalsoofUUII}feP _m.dZ"궲Ofucourse,onecannothopGetogetuZK(I)M6=;ifKisunotalgebraicallyclosed,sinceforinstanceZQIJ(x^2^+{1)n=;.Moreover,althoughthequestionofoRwhetheroRZLt(I)h=;oroRnotdoGesnotdependonwhichalgebraicallyclosed eld_|LKwe_|choGose,theset_|ZLt(I)itselfclearlydoGes.F*orinstance,if궵I=(y8x^2|s)Q[x;y[ٲ],UUthenUU(;^2L)2ZC5(I),UUbutUU([;^2L)㊵=2 8Z;1404`2.Gr`obnerTBases-ō>;De nitionT2.6.15.o3|Letk*K|LbGek*a eldextension,andletk*S L^nq~.Then thesetofallpGolynomialsеf2K[x1|s;:::;xnq~]suchthatf(a1;:::;anq~)=0forfallpGointsf(a1|s;:::;anq~)Fڸ2Sformsfanidealofthepolynomialring궵K[x1|s;:::;xnq~].Thisidealiscalledthev\ranishingeidealofSHinK[x1|s;:::;xnq~]andUUdenotedbyUUI(S)."Using=thisnotation,thestrongversionofHilbGert'sNullstellensatzsaysthattheopGeration׸I(::: )isaninverseto׸Z_(:::)ifoneconsidersonlyradicalidealsUUinpGolynomialringsoverUUalgebraicallyclosed elds.TheoremT2.6.16.jUF(HilbQert'sTNullstellensatz)L}'etdKbedanalgebr}'aicallydclosed eld,andletdI-}beaproperidealof궵K[x1|s;:::;xnq~].Then A;vI(Z_(I))= p o fe.8qIPr}'oof.6T*oIshowtheinclusionII(Z_(I))p Yfe.8wvIM,IsuppGosethatapolynomial궵fJ27,P=K[x1|s;:::;xnq~]satis esf^i 27,Iforsomei0.Thenwehave궵f^ig۲(a1|s;:::;anq~)=0foreverypGoint(a1|s;:::;anq~)2Z_(I).Thuswealsohave궵f(a1|s;:::;anq~)=0UUforUUeverypGointUU(a1|s;:::;anq~)2Z_(I),UUi.e.fڧ2I(Z_(I))."T*oprovetheotherinclusion,wemayassumethatI6=(0).W*echoGose궵f퍸2I(Z_(I))ָnf0gFandFasystemofgeneratorsfg1|s;:::;gsF:gofI.Letxn+1궲bGe.anewindeterminate,andconsidertheideal.I^0^3=IPc[xn+1]m+(xn+1f1)inforsomemr0 QJandUUsuitablepGolynomials\qo~UUh1;:::;\q~hsz2Pc,whichmeansthatUUfڧ2p ofe.8wvIJ.C"궲Our& nalresultinthissectionprovidesareformulationofHilbGert'sNull-stellensatzUUwhichwillproveusefulinthe nalsectionofthisbGook.CorollaryT2.6.17.lL}'et߃Kbe߃a eld,letIeb}'eaproperidealinthepoly-nomialƥringƥPVV=ǵK[x1|s;:::;xnq~],let}fe 5UKŸb}'ethealgebraicclosureofƥK,let궟}feP ݻ=#}fe 5UKY=[x1|s;:::;xnq~],andletĵfSb}'eapolynomialinĵPc.IffSb}'elongstoallmaximalide}'alscontainingI}feP ,thenfڧ2p ofe.8wvIJ. :$Wō>;)2.6HilbAert'sTNullstellensatz9141-ō>;Pr}'oof.6FirstweobservethatPropGosition2.6.11impliesʵfڧ2I(Z_(I}feP)).This =ideal/equals/p fe U wsI}feP byHilbGert'sNullstellensatz2.6.16.Thereforethereexists aTnumbGerTij2NsuchTthatf^iOE2I}feP .TTheclaimnowfollowsfromPropGosi-tionUU2.6.12.:2%Exercise1.Giv9e3FadirectproAofforthefactthatconditionb)ofTheo- %remT2.6.6impliesconditiona)ofthattheorem.%Exercisew2.}LetkKSbAeka eldandf(x):2K[x]ankirreduciblepAoly-%nomial.FindamaximalidealminK[x;yR]whic9hcontainstheideal%IF!=(f(x);f(yR)),andcomputethein9tersectionofmwithK[x]andK[yR].%ExerciseT3.UU(StructureofMaximalIdealsinR[x1*;::: ;xn7])(LetTmbAeTamaximalidealinTR[x1*;::: ;xn7]. *?a)7xLet4n.=1.4Sho9wthatmiseithergeneratedb9yapAolynomialoftypAe7xx1Tawitha2R,orapAolynomialoft9ypex-=2h1+Tax1+bwitha;b2R7xandTa-=2884b<0.7xHint:Usethefactthatifa+ibisacomplexzeroofapAolynomialin7xR[x],GthenalsoGaibisGazero,toc9haracterizeirreduciblepAolynomials7xinTR[x].)b)7xLetnnK=2andf1=x-=2h1+ma1*x1+b1*,nf2=Kx-=2h2+a2*x2+b2 Kwith7xa1*;b1;a2;b2m2Randa-=2h1t4b1<0,a-=2h2t4b2<0.Sho9wthattheideal7xIF!=(f1*;f2)TisTnotmaximalinR[x1*;x2].7xHint:TUsethefactthatTR[x1*;x2]=(f1)isTisomorphictoC[x2*].*6c)7xLet<nY=2and<assumethatx-=2h1R+( a1*x1+b12Ym<witha1*;b12YRand7xa-=2h1I4b1J<p0.Sho9wthatthereexista2*;b22pRsuc9hthatwehave7xm=(x-=2h18+8a1*x1+b1*;x2a2*x1b2*).)d)7xIn#thegeneralcasepro9ve#thefollo9wingfact:eitherthereexistnumbAers7xa1*;::: ;an~2GR#nsuc9h#nthatm=(x1BFa1*;::: ;xn5an7)#nor,#nuptoapAer-7xm9utation[oftheindeterminates,thereexist[a1*;b1;a2;b2Q::: ;an7;bn2R7xsuc9hP8thatP8a-=2h14b1m<0andm=(x-=2h1+a1*x1+b1*;x2a2*x1b2*;::: ;xn77xan7x188bn).%Exercise~4._LetEf2nQ[x;yR]bAeEanon-constan9tpolynomial.Pro9veEthat%Z-O+msbm6Q*(f)Z(f).%Exercise5.>6LetK5LbAea eldextension,letPک=K[x1*;::: ;xn7],let%fw;f1*;::: ;fs2PH,TandletTIF!=(f1*;::: ;fs). *?a)7xSho9wTthatTZL(f)=ZL((f)).)b)7xSho9wTthatTZL(I[)=\-=s~ƍi=1 sZL(fi,r). 덍*6c)7xSho9wTthatTZL(I[)=ZL(`pR`aHv[I t).%Exercise6. h!Let'KLbAe'a eldextension,let'I9beanidealin%K[x1*;::: ;xn7],TandletTSYbAeasubsetofL-=n7. *?a)7xSho9wTthatTI0(ZL(I[))I[.)b)7xSho9wTthatTZL(I0(S))S. OWō>;1424`2.Gr`obnerTBases-ō>;%Exerciseu87.vNLetR.bAearing,andletIJKandJq~bAeidealsinR>.Pro9ve %theTfollo9wingrules.j*?a)8x&pA&aH tȟ #ڍ`pR`aHv[IP=Ɵ`p G`aHv[I 덍)b)8x`p@-ڟ`aHџ[I“\8JWq=Ɵ`p G`aH [I[J=Ɵ`p G`aHv[I\8`p Ê`aHC[J 6*6c)8x,Zp@-ڟ,ZaHӦI[jiJ=Ɵ`p G`aHv[IforTallTi1.)d)7xIfTIɯisTanin9tersectionofprimeideals,then`p ʦ`aHv[I=I[. &j*Se)8x&pA&aH%l #ڍ`pR`aHv[I+8`p Ê`aHC[Ji=ƟOp GOaH4I“+8J%ExercisetR8.GLetMKk5bAeManalgebraicallyclosed eldandI{apropAerideal%ofP!=>K[x1*;::: ;xn7].Inthisexercisew9eusetheZariskitopAologyonK-=n%de nedTinT:utorial27.%TheidealIJissaidtobAereducibleifitisthein9tersectionoftwostrictly%biggerqideals.AqhclosedsetofatopAologicalspaceissaidtobereducible%ifTitistheunionoft9woTpropAerlycon9tainedclosedsubsets. *?a)7xSho9wTthatifTIɯisreducible,thenZ(I[)isreducible.)b)7xGiv9eTanexamplewhichshowsthattheconverseisnottrue.*6c)7xSho9wTthattheconverseofa)istrueifTIɯisaradicalideal.)d)7xLetrGI&bAerGaradicalideal.Pro9verGthatthefollo9wingconditionsareequiv-7xalen9t.<1)I?qThereW*existt9woW*idealsW*I1*;I2+P suchthatW*I^=+I1n\I2 andI?qI18+8I2m=PH.<2)J?qZ(I[)isdisconnected,i.e.itistheunionoft9wodisjointclosedI?qsets.%Exercise+9.pLetaKbAeaanalgebraicallyclosed eld,andletIbbAeaproper%radicalTidealofTPک=K[x1*;::: ;xn7].*?a)7xPro9veTthatTZ(I[)is niteifandonlyifIɯisoftheformIF!=Ɵ?T= Gsƍ Gi=1Le(x188ai1;::: ;xn,oainʩ)7xwithTpairwisedi eren9tpAointsT(a11N;::: ;a1n);:::;(as1n;:::;asn)2K-=n.)b)7xSho9w֡thatifa)holds,then֡ZL(I[)=Z(I)֡for֡ev9eryextension eld7xLK. d=Wō>;)2.6HilbAert'sTNullstellensatz9143-ō>;T utorialT26:GraphColourings2궲SuppGose(wearegiven3di erentcoloursandagraph(havingnnoGdes(and atUUmostonearchbGetweenanytwonoGdes,e.g.<ۍ[R~8g@$]g@F?XF?8۟$]e$]~1[%]2[U*3XU*4b{5)%]6 g%]7N9..oS..nDm.y\.'J.8.&:.1x...H.:ޟ0.̟nb..D|.' .d.Nr#.`߰.N=.X<Z.*W..bq.P..kП. .ǬG2.u.#L.v.d=f.-Rz.@.. .7 3. q'. .@ A.ԟ ).Ÿ g[.J . u. .Tz ].h .V ة.^D 6. 2 S. 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ndoutifthenoGdescanbecolouredinsuchawaythatno4archconnectstwonoGdesofthesamecolour.InordertousethetheoryofGrobnerUUbasestosolvethisproblem,weintroGducethefollowingnotation."The%8colourswillbGecalled%81,0,%8and1.%8TheywillbGeidenti edwiththeelementsofthe eldF3t=)Z=(3).F*ori)=1;:::;n,wechoGoseanindetermi-natej/xi{andj/formthepGolynomialringPMh=F3|s[x1;:::;xnq~].j/W*eshallidentifya:colouringofthegraphwithapGointof:F^nl3 suchthatthe:i^th coGordinateoftheUUpGointcorrespondstothecolouroftheUUi^th `node.a)%Showthatthesetofzerosoftheideal(x^3l11px1|s;:::;x^3፴n&xnq~)isprecisely%theUUsetofallcolourings.?b)%ProveQthattheQi^th (andj^th noGdeQofthegraphhaveQdi erentcoloursif%andConlyifthecolouringisazeroofthepGolynomialCx^2;Zi+AxiTLxjJ+x^2;Zj1.\Ic)%InFaddition,wemayassumethatthe rstandsecondnoGdesareconnected,%thatPthe rstnoGdehascolour\0",andthatthesecondnodehascolour%\1".uWhatpGolynomialequationsdoesthisimplyforthecolouringsunder%consideration??d)%W*ritejaC0oLCoA!wprogramjColouring3(::: )jwhichtakesalistofpairsfrom%f1;:::;ng^2representingEthearchesandcomputesanidealEIP-whose%zerosarepreciselythecolouringsofthegraphrepresentedbythosepairs%whichUUsatisfyouradditionalconditions.\Ie)%ApplyyourfunctionColouring2(::: )tothegraphabGove.ThenuseC0oLCoA%to+computethereduced+LexͲ-Grobnerbasisofthisideal.DoGesthegraph%have'acolouringofthedesiredkind?Ifyes,howmanydi erentones?%(Hint::UseHilbGert'sNullstellensatztointerprettheanswerofyourcal-%culation.)f)%Considerڴthegraphformedbyconnectingthecenterofaregular7-gon%toaitsvertices.(Ithas8noGdesand14arches.)UseC0oLCoA!andtheW*eak%NullstellensatzTtoshowthatthisgraphcannotbGecolouredasrequired%abGove. pWō>;1444`2.Gr`obnerTBases-ō>;T utorialT27:AneVarieties궲Let3KNh2LbGe3a eldextension.F*oreveryideal3IinK[x1|s;:::;xnq~]we consider1theset1ZLt(I)ܸL^n as1giveninDe nition2.6.10.F*orthemoment,letuscallasubsetofԵL^n ^Razero-setifitisoftheformZLt(I)L^n ^RforsomeidealUUIK[x1|s;:::;xnq~].a)%ProveUUthefollowingclaims.)1)8x;UUisUUazero-set. )2)8xL^n ӲisUUazero-set.)3)7xIfUUE1|s;:::;EsareUUzero-sets,then[^s;Zi=1 tOEiisazero-set.)4)7xIf4vJ*is4vasetofindicesandfEj6gjg2Jaasetofzero-setsindexedbyJ9,7xthenUU\jg2JEjisUUazero-set.%Deducethatthezero-setsofL^n ^canbGetakenastheclosedsetsofa%topGology*,&whichwedenoteby&T*opZqƴKI;L%of>A^nbK containingS҄hasauniqueminimalelement(withrespGectto%inclusion).UUThiszero-setiscalledtheZariskiTclosureofUUS.?h)%Let}K47bGe}analgebraicallyclosed eld,andletS bA^nbK.Prove}thatthe%ZariskiUUclosureofUUSisgivenbyUUZ_(I(S)). 붟Le @궄jbRō>;-ō>;궾3.%NFirstffApplicationsvNRÎipenessNhowfundamentalthenotionofasyzygyis,forinstancebGecauseitplaysUUanessentialroleintheconstructionofBuchbGerger'sAlgorithm."However,theimpGortanceofsyzygiesinAlgebragoesfarbeyondwhatwehave@seenuptonow.ThereforeitishighlyrelevqanttobGeabletocomputethem.Infact,our rstachievementinthischapteristhesolutionoftheproblemofcomputingSyzE8qƴP'(g1|s;:::;gsF:)forarbitraryvectorsg1|s;:::;gs R2Pc^r&,whereP#6=K[x1|s;:::;xnq~]isapGolynomialringovera eldK;/(seeTheo-remg3.1.8).Exploitingthisnewability*,wewillthendiscoverwaystoexplic-itly_pGerformbasicoperationsamongidealsandmodulessuchasintersections,colonUUideals,andcolonmoGdules(seeSection3.2)."Next,weenjoyatripintotherealmsofComputationalLinearAlgebraandoComputationalHomologicalAlgebra.Usingsyzygycomputations,wecani- ndpresentationsforthekernelandtheimageofalinearmapbGetween nitelyhgeneratedhӵPc-moGdules,andwecanliftalinearmapalonganotherone.uEvenmorechallenging,butwithinourgrasp,isthetaskofcomputingpresentationsofHom-moGdules.Usingtheingredientsgatheredearlier,weshallUUconcoGctanalgorithminSubsection3.3.B."Havingsafelyputinstoremanyfruitsofsyzygycalculations,wemovetothenextorchard.Eliminationtheoryprovidesuswithaparticularlyfertilesoilforfurtherapplications.Thisisafascinatingsub8jectwhoseroGotsliein 7Wō>;1464`3.FirstTApplications-ō>;궲classicalGFgeometry*,whereitisrelatedtopro8jections.FromthepGointofview ofComputationalCommutativeAlgebra,wehavetopGerformanimportantswitch.F*romhereonwearenolongerallowedtouseanarbitrarymoGduletermordering.Instead,wehavetolimitourselvestothemorerestrictiveclassofUUeliminationorderings."After{weexplainthemaintheoremforcomputingeliminationmoGdulesinSection_53.4,manyotherripGeningfruitswillbecomereadyforpicking.UsingthemethoGdoftagvqariables,weobtainnewwaystopGerformthebasicopera-tionsonmoGdules.Then,inSection3.5,welearnhowtocomputesaturationsandUUhowtocheckradicalmembGership."Other!impGortantapplicationsarecollectedinSection3.6,wherewedis-cussringhomomorphisms.Amongotherthings,weshowhowto ndpresen-tationsܛforthekernelandtheimageofahomomorphismof nitelygeneratedalgebras,howtosolvetheimplicitizationproblem,howtocomputemini-malk}pGolynomialsofelementsinanealgebras,howtocheckmembGershipin nitely6generatedsubalgebras,andhowtoanalyzesurjectiveandbijectivehomomorphismsUUbGetweenpGolynomialrings."The nalSection3.7ofthischapter,andhenceofthisvolume,representsthequltimateactofharvesting.Itisdevotedtotheproblemofsolvingsystemsof :pGolynomialequationse ectively*.Atthatstageofyourreading,youwillbGe=challengedtorecallalmostalltheknowledgethatyougatheredduringthe׬journey*,tobGecomeaware׬oftheskillsandthetoolswhichyouhavelearned,andtousethemtodigoutther}'ootsѲofsystemsofequations.Ourlastv eldofinvestigationvalsocontainsalgorithmsforcheckingwhetherthesetcofsolutionsofasystemofequationsis nite,forcomputingsquarefreepartsUUofpGolynomials,andfor ndingradicalsofzero-dimensionalideals."Andwhatthen?TherearecountlessotherapplicationsofGrobnerbases,and1newonesarediscovered1almostdaily*.Butsincewewantedto nishthisbGook5beforethenewmillennium,wedecidedtostophere.OtherapplicationsandUUinterestingtopicswillbGecontainedinV*olume2.So,seeyoulater!"⍑cSomeWordsAb`outNotationLSuchN;L23.FirstTApplications9147-ō>;궲itTwouldbGeconvenienttoidentifyѲtheminanaturalway*.ThekeypGointis thatUUwhenweloGokatUUG^=(g1|s;:::;gsF:),weseethatitisalreadywrittenasar}'ow.WThuswearenaturallyleadtothinkofthevectorsWg1|s;:::;gs"asc}'olumnve}'ctors,UUandtoidentifyUUG&withthematrixhavingthosecolumns."Indeed,vfromnowonweshallmakethisidenti cation.ItallowsustointerpretSanexpressionSPލʴs%i=1fiTLgi۲astheresultofamatrixmultiplicationintheUUfollowingway*.qԍs {cX t|"i=1cfiTLgid=(g1|s;:::;gsF:)X0Bfi@ Qf1b.b.b. 0nfsX!ë1!Cfi!A! 궲HereȖtherowȖ(g1|s;:::;gsF:)isȖwrittenasatuple,withtheappropriatecommas,and58interpretedasamatrix.Nowrecallthatasyzygyof58G hasbGeende nedasS>atupleofpGolynomialsS>(f1|s;:::;fsF:)nF2Pc^s suchS>thatPލys%yi=1pfiTLgi’=0.S>So,givenTatupleofsyzygiesTSPofGѲ,TwecanreadtheformulaTGByS=0asTthecorrespGondingUUmatrixexpression."Onttheotherhand,givenamatrixtMoftsizerwkt,wecanspGeakaboutthe궵Pc-submoGduleofPc^r T,generatedbythetvectorsrepresentedbythecolumnsofI۸M.IConsequently*,whenwewriteISyz(M),wemeanthemoGduleofsyzygiesofthevectorsrepresentedbythecolumnsofM.F*oratupleofsyzygiesS궲ofUUM,UUthecorrespGondingmatrixexpressionisagainMS=0."Finally*,amap:Pc^s*!Pc^r R8de nedbysendingeiStothevectorgiSfor궵i=1;:::;sBAcanBAbGerepresentedbytheBAs-tupleG^=(g1|s;:::;gsF:)asBAwellasbytheFqmatrixwhosecolumnsrepresentthevectorsinFqGѲ.AndnowitshouldbGeclearUUthatthismatrixcansafelybGecalledUUG&again."TheupshotofthisdiscussionisthatwerepresentalinearmapbGetweenfreemoGdulesbythematrixwhosec}'olumnsaretheimagesofthecanonicalbasis_Tvectors,thatthegeneratorsofthesyzygymoGduleofatupleofvectorsare{thec}'olumnsMofthesyzygymatrix,andthatthesyzygymatrixisontheright-handsideUUofthecorrespGondingmatrixproduct. Wō>;1484`3.FirstTApplications-ō>;3.1**ComputationofSyzygyMo`dulesdXzTheNinthepreviouschapter,welet^>KZbGea eld,P*=K[x1|s;:::;xnq~]apGolynomial=Pring,=PM`ԸIPc^r anon-zeroPc-submoGdule,andGኲ=I(g1|s;:::;gsF:)atupleofnon-zerovectorswhichgenerateҵM.F*urthermore,weletҵbGeamoGdulektermorderingonkT^nq~he1|s;:::;ermi,andwewriteLM18Dz(giTL)=citie i궲withw#ciS2pK,ti2pT^nq~,w#and i2pf1;:::;rGgfori=1;:::;s.w#IfwedenotethecanonicalbasisofthePc-moGdulePc^s byf"1|s;:::;"sF:g,thes-tupleGcorre-spGonds Dtothesurjective DPc-linearmap:Pc^spW!M7_given Dby("iTL)=gitfor궵i=1;:::;s. RecallthatthesyzygymoGdule SyzT(GѲ) isnothingbutthekernelofUU. (Wō>;B+3.1ComputationTofSyzygyMoAdules9149-ō>;"궲Our( rstgoalistointroGduceacertainmoduletermorderingonthesetof terms>of>Pc^s suchthatthemapissomehow\compatible"withthemoGduletermUUorderingsonUUPc^s andPc^r1.ҍDe nitionT3.1.1.is~OnT^nq~h"1|s;:::;"sF:i,wede neacompleterelation츲inthefollowing#way*.Let#t"i oandt^09"j ϲbGetwoelementsof#T^nq~h"1|s;:::;"sF:i,where궵t;t^0Q2T^n ӲandUUi;jY2f1;:::;sg."Thenwelett"itt^09"jIJifwehaveL*T.=(tgiTL) >IL*T&؟n(t^09gj6),orifwehaveL*T E%۲(tgiTL)=L*TW(t^09gj6)andiȸj.Therelation׻iscalledtheorderinginducedTb9yT(͠;GѲ)onT^nq~h"1|s;:::;"sF:i."UsingYDe nition2.3.4andPropGosition2.3.5,wecanrephrasethede -nitioniofiA0bysayingthatit"il. t^09"j VifdegBqƴ;G h(t"iTL)> zxdegqƴ;G%)w(t^0"j6),iorifdeg#Eqƴ;G/(t"iTL)L=deg۟qƴ;G K(t^09"j6)andiLj.NowwecheckthatthisrelationisindeedUUamoGduletermordering.LemmaT3.1.2.\Ther}'elationNde nedaboveisamoduletermorderingonT^nq~h"1|s;:::;"sF:i.Pr}'oof.6AccordingѰtoDe nition1.4.15wehavetocheckthatѰwisre ex-ive,9*antisymmetric,transitive,compatiblewiththemonomoGdulestructure,andthatitde nesatermordering.AlltheproGofsarestraightforward;here$Yweprovethetransitivity*.Let$Yt;t^09;t^00 82T^n ײandi;jR;k]2f1;:::;sg궲suchthatwehavet"i  t^09"j xt^00r"k됲.F*romthede nitionofӾwegetgL*T Km(tgiTL); L*TA:в(t^09gj6) L*TA:(t^00rgk됲).gF*urthermore,weeitherhaveL*T E%۲(tgiTL)@>ML*TFܟr(t^00rgk됲),NorwehaveNL*TAݟs(tgiTL)@=L*TFFܲ(t^09gj6)=L*TF(t^00rgk됲)Nand궵ijYkP.UUBothtimesweendupwithUUt"idMt^00r"k됲.e@]t"궲Whatlisl3goGodfor?IfthevectorsinlGG=forma[ٲ-GrobnerbasisofM,we˸cantrytocompute˸Syz(GѲ)˸viathefollowingsteps:fromTheorem2.3.7weknowanexplicitsystemofgeneratorsofSyz (LMjC(GѲ)),byConditionD1|s)those3syzygieslifttosyzygiesof3GѲ,andbyPropGosition2.3.11thoseliftingsgenerate thedesiredsyzygymoGdule.SothemaintaskistomaketheprocessofUUliftingmoreexplicit."LetusrecallsomenotationfromChapter2.F*ori;j2tf1;:::;sg,we {de nedtij =텍lcmj(ti*;tja)ŭfe! ⦴ti-&andij= gѱ1&feٟci *tij "i 1ך&feOBcj Ztjgi"j6.F*urthermore,wede ne 뮍B=f(i;j)j1i;1504`3.FirstTApplications-ō>;궲that^L*T"(zp)uθ2hL*T IJ(ij )j(i;j)2Bi.^Thede nitionof^%impliesthat L*T E%z(zp)=L*Tjiܲ(z^0>в).gThuswemayassumethatgzNishomogeneouswithrespGecttotheT^nq~he1|s;:::;ermi-gradingonPc^s [de nedinPropGosition2.3.3.ThismeansthatOifwewriteOz= PލHs%Hi=1?c^0;ZiTLt^0;Zi"irwithc^0;ZijY2 Kkandt^0;Zi2 T^nq~,Othenwehave궵t^0;ZiL*T(giTL)=t^0;ZjTL*Ty(gj6)UUforUUall1i)L*T=/Ӳ((ij)).Now>wede ne>sij =ij 6if(ij )=0andsij =ij Pލ $Hs% $Hk+B=1ڃfijgk v"kotherwise.Hb)%TheAsetAfsij j(i;j)2BgisAa!-Grobnerb}'asisofSyz`(GѲ).Inp}'articular,%itisasystemofgener}'atorsofSyz(GѲ).Hc)%L}'etB^0QBbesuchthat˸fij j(i;j)2B^09ggener}'atesSyzJ(LMjC(GѲ)).Then%thesetfsij j(i;j)2B^09gisasystemofgener}'atorsofSyz(GѲ).Pr}'oof.6T*oprovea),werecallthatŵij isahomogeneouselementofŵPc^s궲ofEĵ[ٲ-degreedegiSqƴ;G ò(ij )W=lcm (tiTL;tj6)e i $byETheorem2.3.7.a.ThereforewegetLFN(ij )pP=ij {H2Syzo(LMjC(GѲ)).Consequently*,if(ij)pP6=0,thenL*T E%۲((ij ))x< ږdeg%qƴ;G%(ij)YbyYPropGosition2.3.6.b,andthedesiredrepre-sentationUUfollowsfromConditionUUA2|s)inTheorem2.4.1."Nextv^weproveb).Ifv^Syz}(GѲ)&=0,v^thenalsov^Syz(LMjC(GѲ))&=0v^byv^Condi-tionْD1|s)ofْTheorem2.4.1,andwehaveْB}=;.ْThuswecanassumethatSyz#1(GѲ)6=0.VF*roma)andthede nitionofVwededucethatL*TGG(sij )=tij"iTL.Nowmwetakeanynon-zeroelementmzހofSyz(GѲ).W*ehavetoshowthatmL*Tx(zp)isکamultipleofoneofthetermsinکfL*T IJ(sij )Oj(i;j)2Bg.کThede ni-tionof]yieldsL*T%Z(zp)=L*Tmhl(LF ;GR(z)).ByPropGosition2.3.6.d,wehaveLF ՟;G-=E(zp)2Syz7(LMjC(GѲ)).UUHencePropGosition3.1.3impliestheclaim."TheUUproGofofc)followsfromProposition2.3.11.dU"궲AsastraightforwardconsequenceoftheprecedingpropGosition,wehavetheUUfollowingalgorithmforcomputingsyzygymoGdulesofGrobnerbases. RנWō>;B+3.1ComputationTofSyzygyMoAdules9151-ō>;CorollaryT3.1.5.f(ComputingTSyzygyMoQdulesofGr@obnerBases) L}'etG=d(g1|s;:::;gsF:)beatupleofnon-zerovectorsinPc^r whichforma궵[-GrobnerG?mb}'asisofM.Considerthefollowingsequenceofinstructions.~o1)%Cr}'eateJamatrixJMoverPewithsr}'owsJandinitiallyzerocolumns.Then%c}'omputethesetBG=f(i;j)j1i;1524`3.FirstTApplications-ō>;"궲AtthispGointweknowhowtocomputeasystemofgeneratorsofthe syzygymoGduleofaGrobnerbasis.NowwebGecomemoreambitiousandwantto*bGeabletocalculatethesyzygymoduleofanarbitrarysystemofgenerators궸fh1|s;:::;htVgUUofM,UUwhereweevenallowzerovectors."ThekeyingredientswillbGethefollowing.UsingtheExtendedBuch-bGerger5Algorithm2.5.11,wecancalculateaGrobnerbasis5fg1|s;:::;gsF:gofM궲togetherFwithrepresentationsFgj =Ua1j3h1G+>Ը(O+>Եatj<ht cforj膲=1;:::;s.F*urthermore,afterwehavecalculatedthisGrobnerbasis,wecanusetheDivisionAlgorithm1.6.4to ndrepresentationshj R=Db1j3g12+'%+'bsjgs궲fors]j6Y=1;:::;t.s]Thuswecanexplicitlycalculatethematricess]Aβ=(aij )andUUB=(bij )requiredUUbythefollowingtheorem."But\bGefore,weremindthereaderthatatupleofvectorscanalsobGeviewedasramatrix,namelythematrixwhosecolumnsconsistofthecoGordinatesoftheUUvectors.TheoremT3.1.8.dH(ComputationTofSyzygyMoQdules)L}'etfh1|s;:::;htVgbeasystemofgeneratorsofaP-submoduleM'"ofPc^r&,let궸H=P(h1|s;:::;htV),letfg1;:::;gsF:gb}'ea[-Grobnerb}'asisofM,andletGb}'e~thetuple~(g1|s;:::;gsF:).F;urthermore,supposewearegivena~tys-matrix궸A=(aij )andansRt-matrix߸BƲ=(bij)overP{nsuchthatG=H^Aand̸HQ=9 GByBM.Finally,letMb}'eamatrixwhosecolumnsgenerate̲Syz(GѲ),and let It b}'ethetGtidentity matrix.Thenthec}'olumnsofthematrix궸N@U=(AMqjIt68ABM۲)gener}'atethemoduleSyz(H).Pr}'oof.6F*romUUG^=HAandH=G^B0weUUgetUHN@U=(HAMqjHQ8HABM۲)=(G^MjHQ8GBM۲)=(0qjHQ8H)=0Hence1thecolumnsof1NnaresyzygiesofH.Conversely*,1ifacolumnvectorv 궲is asyzygyof H,wehave G^B@v"=Hv=0. Hencewehave B@v"2Syz7(GѲ),andthereforethisvectorliesinthecolumnspaceofM.F*romtheidentity궵v"=A(Bv[ٲ)+(ItjABM۲)vTȲwemaythenconcludethatvTȲliesinthecolumnspaceUUofUUNy=.KCorollaryT3.1.9.f(ExplicitTMem9bQership)WiththesameassumptionsasinThe}'orem3.1.8,letml=Pލ9s%9i=1XfiTLgi2M,wher}'ehfiŸ2GyPˤfori=1;:::;s,handlethFfNb}'ethematrixconsistingofonec}'olumnwhoseentriesaref1|s;:::;fsF:.~oa)%TheKve}'qualityKvm=H,T(AF9)pr}'ovidesKvanexplicitexpressionofKvmasKva%c}'ombinationofthegivengeneratorsh1|s;:::;ht=ofM.Hb)%L}'etKN>=(AMjItC` ABM۲).KTheneveryexplicitexpressionofKmas%ayc}'ombinationofthegivengeneratorsyh1|s;:::;ht ofMisyoftheform%mk=H7!(AFZ+[θNP})forasuitablematrixPc}'onsistingofonecolumn%ofp}'olynomials. WWō>;B+3.1ComputationTofSyzygyMoAdules9153-ō>;Pr}'oof.6T*oprovea),itsucestocombinethetwoequalitiesmMc=G^FsGand 궸G^=HA.VNowweproveb).IfVm=HQwithVat1-matrixQofVpGolynomi-als,wededucefroma)thatHQN$H(AF9)=0.HenceQN$AF/isasyzygyofOH.OF*romTheorem3.1.8wededucethatQvAb F`F=NJP̲forOasuitablecolumnCmatrixCP}.NowwecombineCmp=HQwithQ=AF{ +|ԸN6PandobtainUUtheclaim."궲Ournextexampleshowshowonecanapplytheprevioustheoreminprac-tice.nfItalsodemonstratesthatthesystemofgeneratorsofnfSyz(H)nfprovidedbyUUthetheoremisingeneralnotminimal.ExampleT3.1.10.h`InExample2.5.7wesawthatfg1|s;g2;g3gwithg1= x^2|s,궵g2f=鞵xy+y[ٟ^2L,andg3=y[ٟ^3 isaGrobnerbasisoftheidealI=(g1|s;g2)in궵Pn= \K[x;y[ٲ]~Kwith~KrespGecttog5= \LexS.W*eletG-= \(g1|s;g2;g3),~Kh1ϲ= \g1|s,~Kand궵h2C=g2|s,UUandwewanttocomputethesyzygymoGduleofUUHβ=(h1|s;h2)."UsingtheExtendedBuchbGergerAlgorithm2.5.11,wecalculatethema-trixA,andusingtheDivisionAlgorithm1.6.4,wecalculatethematrixBM۲.W*eUU nd L.A=^ 41606ĵy 4061)8x8+yJr^gzandB=Z0 @ 1110 1011 1010Z"l1 "lAqҍ"궲AnȻapplicationofCorollary3.1.5nowyieldsthesystemofgenerators궵s12 ?=y["1$C+(x+y)"2$C"3|s, ̵s13 ?=y^3L"1$Cx^2|s"3|s, and ̵s23 ?=y^2L"2$C+(xy)"3궲ofUUtheUUPc-moGduleSyzt(GѲ).Thereforeweget" qOM=Z0 @ My:Isy[ٟ^3_#0 1x8+y<(0\ӵy[ٟ^2K:16+Ҹx^2R&x8yZr81 r8A"궲anda⏸N@U=^ 406y[ٲ(x^2S8y^2L)YCy[ٲ(x8+y)j0l0 40 x^2|s(x8y[ٲ)jx^2j0l0z^qύ궲EvenifwedeletethezerocolumnsinNy=,westillhavenominimalsystemof)generatorsof)SyzH(H),sincethesecondcolumnisamultipleofthethird.Altogether,UUwe ndhSyzx(H)=hy[ٲ(x8+y)"1S+x2|s"2iPc2"궲It=isapparentthatintheprecedingexampleweactuallycouldhavedone withoutMusingtheExtendedBuchbGergerMAlgorithmortheDivisionAlgo-rithm,wsincetheGrobnerbasiswfg1|s;g2;g3gwcontainedwthesystemofgenerators궸fh1|s;h2gwhosesyzygymoGdulewewantedtocompute.ThishappGensquiteoftenifwestartwithanarbitrarysystemofgeneratorsofM5anddetermineauGrobnerbasisfromitbyusingBuchbGerger'sAlgorithm.So,letusstudythisUUcase. Wō>;1544`3.FirstTApplications-ō>;CorollaryT3.1.11.lSupp}'oseC,that,inthesituationofTheorem3.1.8,thema- trixAisoftheformA =(ItjqC{))withat:ܸ(st)-matrixC+overPc.L}'et궸M9Jb}'e9Jamatrixwhosecolumnsgenerate9JSyzi(GѲ).Ifwedecomposeitinthe ;formzMUp=( MmMr033&feNM00e)withzamatrixM^0havingtr}'owsandamatrixM^00 hav-ing΍sc_tr}'ows,΍thenthesyzygymodule΍Syz(H)΍isgeneratedbythecolumnsofthematrixM^0+8C7M^00r.jZPr}'oof.6ByEassumption,thematrixEBisoftheformB=(h33It33ʉfeɱ0 >j).Thereforeweobtain#ItG]ABl=иItIt&=0,#andtheright-handpartofthematrix#N궲inthetheoremcontributesnothingtoSyz)(H).Theleft-handpartofN[9isgivenUUbyUUAM=(ItjqC{))8( MmMr033&feNM00e)=M^0+8C7M^00r.k6"궲AsanapplicationofTheorem3.1.8,wehavethefollowingmethoGdtodetermineYoanirredundantsystemofgeneratorsofaYoPc-submoGduleM Pc^r1,i.e.casystemofgeneratorssuchthatnopropGersubsetofitgeneratescM.NoticethatthesecondconditioninthefollowingcorollarycanbGecheckede ectivelyUUusingtheSubmoGduleMembGershipT*est2.4.10.a.jZCorollaryT3.1.12.lL}'et *fh1|s;:::;htVgbe *asystemofgeneratorsofa *Pc-sub-mo}'duleMwsPc^r1,letH)=(h1|s;:::;htV),andletNb}'eamatrixoverPwhoseXc}'olumnsgeneratetheXPc-moduleSyz(H).XF;oreveryXi2f1;:::;tg,Xthefollowingc}'onditionsareequivalent.~oa)%Wehavehid2hh1|s;:::;hi1 ;hi+1 tO;:::;htVi.Hb)%Theide}'algeneratedbythei^th rowofN $istheunitidealofPc."InLp}'articular,arepeatedapplicationofthisequivalenceallowsusto ndanVirr}'edundantsystemofgeneratorsofVMqwhichiscontainedinVfh1|s;:::;htVg.jZPr}'oof.6BothconditionsareequivqalenttotheconditionthatthereexistsacolumnUUinthecolumnspaceofUUNΒwhosei^th `entryUUis1.OЙExampleT3.1.13.h`Let&˵h1C=x^2|sy[ٟ^2Ͳ1,h2=x^4|sy[ٟ^4Ͳ2x^2y[ٟ^2+͵xy[zp^2׸ 1&fes2>zp^4+ 1&fes2 >,궵h3 ==)xy[zp^2V 1&fes2 ̍zp^4 1&fes2 ̍,andh4=)xyōizp^2 bGepolynomialsinQ[x;y[;zp],andlet斵IxbGetheidealgeneratedbyfh1|s;h2;h3;h4g.UsingthemethoGdde-scribGedinthecorollary*,wetrytoshortenthissystemofgenerators.SinceSyz#1(h1|s;h2;h3;h4)l=h(1;0;2;xyt+@zp^2 );(0;0;2xy+@2zp^2 ;2xy[zp^2-zp^41);궲(x^2|sy[ٟ^2L1;1;1;0)i,wecandeleteh1 yfromthesystemofgeneratorsofI.Since SyzQ7(h2|s;h3;h4)=h(0;2xyA+h2zp^2 ;2xy[zp^2rzp^41);(1;x^2|sy[ٟ^2z ՛1՛&fes2Axy[zp^2 1&fes2Mzp^4;+211;x^3|sy[ٟ^3 }+ ed1ed&fes4 zp^6+ ed3ed&fes2xy  ed3ed&fes4zp^2 )i,QwecanthendeleteQh2|s.Theremainingsystemofgeneratorsfh3|s;h4gofIDzisirredundant,bGecausewecancheckthatSyz#1(h3|s;h4)=h(2xy+82zp^2 ;2xy[zp^2%zp^41)i."This(examplealsoshowsthatwecanshortensomesystemsofgeneratorsinrdi erentways.F*orinstance,wecouldhavedeletedrh2 andthenh3 inorderUUtogettheirredundantsystemofgeneratorsUUfh1|s;h4gofI. ֠Wō>;B+3.1ComputationTofSyzygyMoAdules9155-ō>;%Exercisep1.i,Letf1*;f2m2P]bAet9wonon-zeropAolynomials.Supposethat %theϥmoAduleϥSyzf(f1*;f2)PH-=2 C/isgeneratedb9yasinglevectorϥ(g1*;g2)2PH-=2.%ThenTsho9wthatTf2?isamultipleofTg1*,andthatgcdN(f1*;f2)=f2*=g1*.%Exerciseo2.,ComputeasetofgeneratorsofSyzh(H )inExample3.1.10,%usingTthemethoAddescribedinCorollary3.1.11.%Exerciseq3.LetkP%=BK[x;yR;zc],klet!bAeatermorderingonT-=3*,let%g1m=yRzc,g2=xzc,g3=xyR,andletG=(g1*;g2;g3).FindasubsetB-=0CB%suc9huthatthecorrespAondingsetu2-=0L=Sfij &2j(i;j)2B-=0gisuasetof%generatorsofSyz7(LMBE(G)),butfsij j=(i;j)2B-=0gisnota-Gr`obner%basisTofTSyz7(G),whereTSistheorderinginducedb9y(;G).%Exercise4.Let')G6v=(g1*;::: ;gs),')wheregi2Pp fori=1;::: ;s.')Pro9ve%thatTSyz7(G)=(0)TifTandonlyifs=1.%Exercise5.fLetG"j=s(g1*;::: ;gs),wheregi2Pfori=1;::: ;sand%g1m=1.ODescribAeanexplicitsetofgeneratorsofOSyz(G)OconsistingofsX-1%elemen9ts.%ExerciseT6.jLetTIɯbAeTanidealinPک=K[x1*;::: ;xn7]. *?a)7xAssume~that~I9isaprincipalidealgeneratedb9yanon-zeroelement~f.7xThenOzsho9wthateveryelementofOzIhasauniquerepresentationasa7xm9ultipleTofTf.)b)7xAssume'rthat'rI=Z(f1*;::: ;fr,p),wherefi2ZPpUfori=1;::: ;ri9and7xr(>71,HandletHf2I[.Thensho9wthatHfCCcanbAerepresentedinmore7xthanTonew9ayTasacom9binationofTf1*;::: ;fr,p.%Exercisev7.8LetNfh1*;::: ;ht\pgbAeNasetofv9ectorsinNPH-=r rwhichgenerates%amoAduleMPH-=ruS,letm2M,andlet9|bAeatermorderingoft9ype%PosToBFoniT-=n7h"1*;::: ;"t+1 i.iExplainho9wonecanusetheknowledgeof%a_R-Gr`obner_basisofthesyzygymoAduleSyz(m;h1*;::: ;ht\p)to_giv9ean%alternativ9eTmethoAdforcomputingexplicitmembAership.=T utorialT28:Splines궲SuppGose`%wehaveaclosedintervqal`%[a;b]R,`%wherea;b2Randa;1564`3.FirstTApplications-ō>;궲bGoundedǀfunctionǀfڧ:[a;b]!R(whichǀmaybGeverycomplicated)forwhich we8know nitelymanyvqalues8did=f(ciTL)fori=0;:::;kP.8InthiscaseweareloGokingyforasplinefunctionys:[a;b] !Rwhichyapproximatesfzwell,yi.e.forUUwhichthenumbGerUUkfLo8sk1?=Vsupqt2[a;b]ϸfjf(t)s(t)jgUUisUUsmall.qa)%TheOsimplestcasesofsplinesaresinglepGolynomialspassingthroughthe%pGointsa(c0|s;d0);:::;(ck됵;dk).aF*oraߵi=0;:::;kP,awede neaߵ`i0J=Q Mǟjg6=i텍txcjZ*ŭfeMci*cj2y.OA%ShowtthattheLagrangein9terpQolationpolynomialt`=Pލ USk% USi=0tJdiTL`ihas%degreeR'kandRpassesthroughthepGoints(c0|s;d0);:::;(ck됵;dk).RW*ritea%C0oLCoADfunctionLagrange.(::: )whichtakesthelistofpairs(ciTL;di)and%computesUUtheLagrangeinterpGolationpolynomial.?b)%T*akethefunctionsfH:[0;2[ٲ] /!Rgivenbyf(t)=sin(t)and%g:H+[2;2] d!RDzgivenbyg[ٲ(t)H+= Ա1{^&fe_1+25t2.Ineachcase,dividetheinter-%vqalintoki=4equalpartsandcomputethecorrespGondingLagrangean%interpGolationUUpolynomial.%Hint: FirstwriteaC0oLCoA 7function Sinv(::: ) whichusestheT*aylorexpan-%sion[tocomputethevqalueof[sinIy(t)[uptoacertainnumbGer[ofdecimal%digits.ThenwriteaC0oLCoA"functionValues$`(::: )whichtakesthetuple%(c0|s;:::;ck됲)mandmthenameofthefunctionandcomputesthelistofpairs%[(c0|s;d0);:::;(ck됵;dk)].UUFinally*,thislistcanbGeusedinUULagrange.U=(::: ).,84o,ffoa3<p8fff2.2.j2.,2.En2.2.2.42.v2.g2.2.B<2.~2.2.2.2."Ÿ.w,c.W1<.P_.g.-k͂>.+͠.k. 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Wō>;1604`3.FirstTApplications-ō>;3.2**ElementaryOp`erationsonModules$W`AÎndN;3.2Elemen9taryTOpAerationsonModules9161-ō>;"궲A}niceconsequenceisthepGossibilityofpresentingamoGduleofthe formԵM=(M2\޵N)viageneratorsandrelations(seeCorollary3.2.6).An-otherniceextrabGonusisthediscoverythatsyzygiesprovideamethoGdforcomputingAgreatestcommondivisorsandleastcommonmultiplesofmulti-vqariatepGolynomials,withouthavingtoresorttofactorizingalgorithms(seeCorollaryUU3.2.9)."CanowedoDivision?ֱSuppGosewearegiventwonon-zeropGolynomials궵g[;h2P_suchthatеg"=fhforsomepGolynomialfڧ2Pc.Thisequationimpliesthat?۵fSjis?ageneratoroftheidealfa2P*ja h2(g[ٲ)g.?F*ortwo?idealsI;JQPc,a$Qnaturalgeneralizationistoconsidertheideal$QI:lP ƵJQ=fa2P*jaظJIg.W*e`callitthec}'olonideal`ٲof`ٵI)byJ9.Thisconstructionisparticularlyusefulinthecomputationoftheso-calledprimarydecompGositions(seeT*utorial43)andvinalgebraicgeometry*,whereitisrelatedtotheproGcessofremovingirreducibleUUcompGonentsfromanalgebraicvqariety*."TheBde nitionofcolonidealscanbGeextendedintwoBwaysintotherealmofmoGdules.ZOnemethoddescribesanoperationontwoZmoduleswhichproducesanv]ideal,calledthec}'olonKideal.v]ItcanalsobGeviewedastheannihilatorofacertainlquotientmoGdule(seeDe nition3.2.10).TheothermethodyieldsanopGerationontwomodulesandanidealwhichproducesamodulecalledthec}'olonmoduleUU(seeDe nition3.2.17)."Ine-thesubsection\ColonIdealsandAnnihilators"weshowtwodi erentapproachessetothecomputationofcolonideals,onebasedonintersectionsandltheotherbasedonsyzygiesofasuitablematrix(seePropGosition3.2.15).And3finthesubsection\ColonMoGdules"weshowtwodi erentapproachesto,thecomputationofcolonmoGdules.Again,oneisbasedonintersectionsandFtheotheronsyzygiesofasuitablematrix(seePropGosition3.2.22).A nalNfapplicationofthetechniquesdevelopGedaboveNfisamethodforcheckingwhether+~agivensequenceofpGolynomialsisaregularsequence,apropertywhichUUweshallreexamineinT*utorial33andinVolume2.7]"LetKbGea eld,n1,P =K[x1|s;:::;xnq~]apGolynomialring,r1,궵 a/GmoGduletermorderingon/GT^nq~he1|s;:::;ermi,andG^=(g1|s;:::;gsF:)2(Pc^r1)^suatuplexQofvectorswhichgenerateaxQPc-submoGduleMlofPc^r1.xQF*urthermore,welet H =@(h1|s;:::;htV)2(Pc^r1)^tbGe atupleofvectorswhichgenerateanother궵Pc-submoGduleUUNlpofPc^r1."F*oranrG^0]H+s^02matrixMwithentriesinPc,weletSyz$(M)bGethesyzygymoGduleofthetupleofthecolumnsofM,eachviewedasavectorinPc^r7r0,asalready&explainedintheintroGductionofthechapter.ThefollowingremarkcollectssomeopGerationsonmoduleswhichcanbecomputedinacompletelytrivialUUway*.hRemarkT3.2.1._uHLetg3M=߸hg1|s;:::;gsF:iandN=߸hh1|s;:::;htVibGeg3twog3Pc-sub-moGdulesofPc^r1asabove,andletI7PbeanidealwhichisgeneratedbyasetUUofpGolynomialsUUff1|s;:::;fu:bgPc.a)%ThesumM+ŵNɲisthePc-submoGduleofPc^r V߲generatedbythesetof%vectorsUUfg1|s;:::;gsF:;h1;:::;htVgPc^r1.(Wō>;1624`3.FirstTApplications-ō>;?ֲb)%TheGproGductGImֵMbisthePc-submoGduleofPc^r xgeneratedbythesetof %vectorsUUffiTLgjĸj1iu;q1jYsgPc^r1.\Ic)%F*orieveryid1,thepGoweriI^d AistheidealofPhiTL.Then ^(a1|s;:::;as+#t o )isinSyz#1(g1|s;:::;gsF:;h1;:::;htV),uandwecan ndpGolynomialsup1;:::;pu7Ǹ2ePysuchthatR`(a1|s;:::;as+#t o )l=Pލ u% jg=1dpj6vj6.R`Inparticular,wegetvȫ=l(a1|s;:::;asF:)= g$궟Pލxu%xjg=1.zHpj6(f1j3;:::;fsj)UUwhichUUprovestheclaim.}#PropQositionT3.2.3.q(In9tersectionTofTwoSubmoQdules)L}'etIM3=hg1|s;:::;gsF:iandN=hh1|s;:::;htVib}'eItwoPc-submodulesIofPc^r&,andletzc:Pc^spW!Pc^r Kb}'ezcthePc-linearzcmapgivenby'("iTL)=giίfori=1;:::;s.~oa)%L}'et((fv1|s;:::;vu:bglPc^s+tbe((asystemofgeneratorsofthe((Pc-module%Syz4(g1|s;:::;gsF:;h1;:::;htV),andletvjIJ=(f1j3;:::;fs+#tBj)withp}'olynomials%f1j3;:::;fs+#tBj&2PvforjY=1;:::;u.Thenwehave0aEܵMO\8N3=(1 t(N))=h s 3 Pti=1 Ofij gidj1jYuiHb)%Considerthefollowingblo}'ckmatrixofsize2r8(r+s+t)n6ڿM=^ 4Ir櫸G2o0 4Ir 007H:ߌ^䎍%wher}'e$1Ir is$1ther苸nrkNidentitymatrix.L}'etfv1|s;:::;vu:bg{Pc^r7+s+tbe$1a%system\eofgener}'atorsof\eSyz(M),andlet\evjh=2M(f1j3;:::;fr7+Zs+#tBj@)with%f1j3;:::;fr7+Zs+#tBj X2PvforjY=1;:::;u.Thenn/s"MO\8N3=h(f1j3;:::;fr7j$N)j1jYui<Wō>;3.2Elemen9taryTOpAerationsonModules9163-ō>;Pr}'oof.6Sinceuclaima)followsimmediatelyfromthelemma,itsucesto show(claimb).Let(w1|s;:::;wr7+s+tzbGethecolumnvectorsof(M.IfweloGokatMthe rstandthelastMڵrcompGonentsoff1j3w1^+)}+)fr7+Zs+#tBjwr7+Zs+#tGj=0,weUUobtain bX+0+Bfi+@'Եf1jꨍ-.-.-.'=fr7jX616Cfi6AB5 =fr7+Z1Bj;g1S8g8fr7+Zs]jags R=fr7+Zs+#1Bj h18g8fr7+Zs+#tBjht#bb궲for|jY=1;:::;u,|andtherefore(f1j3;:::;fr7j$N)2MK\0N.|Conversely*,ifwestartwithJ#anelementJ#v"2M9\"N,andifwewriteJ#v"=(a1|s;:::;arm)withJ#pGolynomials궵a1|s;:::;ar42Pc,UUthentherearepGolynomialsUUar7+Z1;:::;ar7+Zs+#tGj2PsuchthatbU:Zv"=ar7+Z1g1S8g8ar7+ZsWlgs R=ar7+Zs+#1woh18g8ar7+Zs+#tRht궲ByUUcombiningthoserepresentationsofUUv[ٲ,wegetthevectorequationva1|sw1S+8g+8ar7+Zs+#tRwr7+Zs+#tGj=0ThusUUtherearepGolynomialsUUp1|s;:::;puz2Psuchthat׭](a1|s;:::;ar7+s+tR)=̴u X tjg=1㉵pj6(f1j3;:::;fr7+Zs+#tBj@)}TheUU rstUUrrcompGonentsofthisequalitynowprovetheclaim.9bUExampleT3.2.4.c;1644`3.FirstTApplications-ō>;De nitionT3.2.5.is~LetBRUղbGeBaring,andletM3=hm1|s;:::;msF:ibGea nitely generated1nRDz-moGdule.1nSupposethatthesyzygymodule1nSyzx(m1|s;:::;msF:)1nhasa nitesystemofgeneratorsfv1|s;:::;vu:bgRǟ^sZ.Moreover,letfe1|s;:::;esF:g궲bGeUUthecanonicalbasisofUURǟ^sVandf"1|s;:::;"u:bgtheUUcanonicalbasisofUURǟ^uN)."W*e$de nean$RDz-linearmap':Rǟ^s! =!M;ֲby'(eiTL)=miyfori=1;:::;s궲andanRDz-linearmap :P-Rǟ^uV!Rǟ^s by [ٲ("j6)=vjAforj⸲=1;:::;u.ThentheUUsequenceNoRǟuꍑ $ _!N Rǟsꍑ8'$7!YMK !00isclearlyexact.Itiscalledapresen9tation1of1MH7viageneratorsandrelations,Vorsimplyapresen9tationofM.Equivqalently*,weshallalsocalltheinducedisomorphism̵MT͍M+3M=֝Rǟ^sZ=hv1|s;:::;vu:biapresentationof̵M.Herethem4residueclassesofthecanonicalbasisvectorsofm4Rǟ^s 5correspGondtothegeneratorsofƵM,andthevectorsv1|s;:::;vu #(generatethemoGduleofrelationsamongUUthosegenerators."GivenktwosubmoGduleskM;NofPc^r&,itisnaturaltoaskforapresentationofnLM=(M I\.N)vianLgeneratorsandrelations.Moregenerally*,wehavethefollowingUUresult.CorollaryT3.2.6.fGivenYthesituationofPr}'opositionY3.2.3,wede neve}'ctors궵wjɲ=V(f1j3;:::;fsj)p*forj訲=1;:::;u.p*Themapp*induc}'esp*aPc-linearp*map궟 feW$β::Pc^s䊸!M=(M~?\g$N).ӿMor}'eover,letӿ ::Pc^u ز(!Pc^s }b}'etheӿPc-linearmapwhichsends"jʓtowjforjY=1;:::;u.Thenthese}'quence@ilPcuꍑXp $ ,'!PcsT΍ʨtW :ግ2 7!M=(MO\8N)0 !00isBapr}'esentationofBM=(M\޵N).Inotherwords,thereisanisomorphismof궵Pc-mo}'dulesM=(MO\8N)T͍+3= UNPc^sɵ=hw1|s;:::;wu:bi.Pr}'oof.6SinceHtheimageofHisM,HitisclearthatH feW$issurjective.ThekernelEofE feW$ isE^1 t(MUݸ\>µN)=^1(N).EByLemma3.2.2,thispreimageequals궸hw1|s;:::;wu:bi=Im( [ٲ).a"궲Ifweneedtocomputetheintersectionofa nitenumbGerofsubmodules궵M1|s;:::;M`ofóPc^r&,ówemayeitherproGceedrecursivelyortrytointersectallsubmoGdulesUUsimultaneously*.PropQositionT3.2.7.q(ComputationTofMultipleIn9tersections)L}'et;`2,;andletM1|s;:::;M`Pc^r lb}'ePc-submodules.;F;oreveryindex궵i2f1;:::;`g,letMi3b}'eamatrixwhosecolumnvectorsgenerateMiTL.~oa)%We5have5M1\51 \51M`=( UO((M1\M2|s)\M3)\ ز)\M`.5Ther}'efore%we~c}'ancompute~M1\ \ M`WbyiterativelyapplyingProposition3.2.3.Hb)%Considertheblo}'ckmatrixiWō>;3.2Elemen9taryTOpAerationsonModules9165-ōLrteM=T0BBBfi@qɍ 1Ir!J8M1A80W(erq07 1Ir' 0<ظM2WG.[u0._.s.s.s....(.(.(.>0.B.F0.WG.[u0._.rq0 1Ir' 0>kZ}0mzظM`T1CCCfiA*Nu%L}'et,fv1|s;:::;vu:bgbe,asystemofgeneratorsofthesyzygymodule,Syz>K(M), %andwriteĵvjIJ=(f1j3;f2j;::: UO)withf1j3;f2j;:::g2PWSforĵjY=1;:::;u.Then%wehavedM1S\8g\8M`=h(f1j3;:::;fr7j$N)j1jYui>Pr}'oof.6TheproGofofthesecondmethodfollowsinthesamewayastheproGofofUUPropGosition3.2.3.b.:ExampleT3.2.8.c"As.canapplicationoftheprecedingtwo.cpropGositions,wecanshowhowtocomputePgreatestcommondivisorsandleastcommonmultiplesofpGolynomi-alsNinNnindeterminates.W*epGointoutthatthefollowingcorollaryprovidesan 'algorithmwhichworksoveranybase eld 'KCoverwhichwecancomputeGrobnerkbases.Inparticular,itdoGesnotrequirefactorizationofmultivqariatepGolynomials."AsTinSection1.2,theexpressionsTgcdV(f1|s;:::;fm)Tresp.lcmV(f1;:::;fm)shallerepresentanygreatestcommondivisorresp.leastcommonmultipleofaUUsetofpGolynomialsUUff1|s;:::;fmgPc.CorollaryT3.2.9.f(ComputationTofgcdandlcm)L}'etm2,letf1|s;:::;fm _2Pc,andletb}'eatermorderingonT^nq~.~oa)%Thehr}'educedh[-Grobnerbasisoftheintersectionidealh(f1|s)\\(fm)%c}'onsists_ofpreciselyoneelement,namelytheelement_lcm(f1|s;:::;fm).%Thusle}'astcommonmultiplescanbecomputedusingProposition3.2.7.|РWō>;1664`3.FirstTApplications-ō>;Hb)%Agr}'eatest5commondivisoroftwopolynomialscanbecomputedviaa)and %theformulaڲgcd(f1|s;f2)=f1|sf2=lcm8(f1;f2).Agr}'eatestcommondivisorof%mor}'ethantwopolynomialscanbecomputedrecursivelyusingtheformula%gcd4(f1|s;:::;fm)=gcd(gcd(f1;:::;fm1);fm).Pr}'oof.6ByKPropGosition1.2.8.a,theidealKIB?=y](f1|s)/\\(fm)isKgeneratedbythepGolynomialfڧ=lcmUS(f1|s;:::;fm).SinceIisaprincipalideal,wehaveL*T E%۲(I)=(L*T %(f)),andthereforethereduced[ٲ-GrobnerbasisofIconsistsofspreciselyonepGolynomial,namelysf.Thisprovessa),andb)isanimmediateconsequenceUUofa)andPropGosition1.2.8.b.u%3.2.B ColonTIdealsandAnnihilators궲NowFwestarttoconsidertheproblemofcomputingcolonidealsandannihi-lators.UUTheyarede nedasfollows.De nitionT3.2.10.o3|LetUURibGeUUaring,andletUlpbGeanRDz-moGdule.a)%GivenUUtwoUURDz-submoGdulesMlpandNofU,UUtheset>ʵN3:lRM=fr52R߸jr8M3Ng%isanidealofRDz.Itiscalledthecolon5Aideal(ortheidealquotien9tif %U3=RDz)UUofUUNlpbyM.?b)%LetE>M\YbGeE>anRDz-module.E>ThesetE>Ann᳟Rq(M)=fr52R߸jr_иM3=0gE>isE>an%idealUUofUURDz.ItiscalledtheannihilatorofM."Colon{idealsandannihilatorsareessentiallythesamething,asournextpropGositionUUshows.PropQositionT3.2.11.wL}'et:ѵRNbe:aring,letUQb}'eanR-mo}'dule,andletMandNb}'etwoR-submodulesofU.ThenvN3:lRM=AnncR(M=(NO\8M))Pr}'oof.6Thede nitionyieldsAnn@3Rϕ(M=(Nϸ\մM))=fr52R߸jrѸM3Nϸ\Mg. SincerxH[M-iscontainedinM-foreveryr52RDz,wegetAnnyR(M=(N_v\H[M))=궸fr52R߸jr8M3Ng,UUandthisprovesUUtheclaim.s"궲OurlgoalistocomputetheabGovelob8jectse ectivelywhenwedealwith nitelyegeneratedmoGdulesovereanealgebras.W*ejustsawthatcomputingcolonidealsisthesamethingascomputingannihilators.ThenextremarksaysY thatforourpurpGosesitsucestocomputeannihilatorsof nitelygen-eratedĵPc-moGdules,whereP-=&K[x1|s;:::;xnq~]isthepGolynomialringovera eldUUK qasUUusual.Wō>;3.2Elemen9taryTOpAerationsonModules9167-ō>;RemarkT3.2.12.e5FLetJزbGeanidealinPc,letMbGea nitelygenerated moGduleovertheane춵K-algebraPV=J9,andlet:n Pћ PV=JbGethecanonicalhomomorphism.W*ecanviewMasa nitelygeneratedPc-moGduleviaUU[ٲ,UUi.e.viafLo8m=[ٲ(f)mforfڧ2Pandm2M."ThentheannihilatorAnn$c:PU=J'(M)istheimageoftheidealAnn$cPR(M)under[ٲ,bGecause[ٲ(f)2Ann~t:PU=J%龲(M)forsomef2Pbmeans[ٲ(f) M=궵fLo8M3=0,UUi.e.itmeansUUfڧ2AnncP|(M)."ThefollowinglemmasolvesourprobleminthecaseofacyclicmoGduleM.LemmaT3.2.13.bG L}'etMò=ohg[ٸiandN=ohh1|s;:::;htVib}'etwoPc-submodules of>Pc^r&,>wher}'eMUiscyclic.L}'etfv1|s;:::;vu:bgPc^t+1Gbe>asystemofgeneratorsofDSyz(g[;h1|s;:::;htV).DWewriteDvjIJ=(f1j3;:::;ft+#1]j)withf1j3;:::;ft+#1]j2PforjY=1;:::;u.ThenRp~N3:lP Ƹhg[ٸi=AnncP|(M=(NO\8M))=(f11x;:::;f1u 6ղ)Pr}'oof.6ItsucestoapplyLemma3.2.2tothemap.:P !Pc^r dزgivenby 17!g[ٲ,UUbGecausewehaveUUN3:lP Ƹhg[ٸi=^1 t(N).ZExampleT3.2.14.h`ConsidertheintersectionI= p14\ p2 ofthetwoprimeidealsMp1C=(y[;zp)andp2=(xѸy[ٟ^2L;UPy[ٟ^3mzp)MinMtheringP*=K[x;y[;zp].MUsingPropGositionj3.2.3,we ndjI=(xy zp;UPy[ٟ^3Yz).jInfact,Corollary3.1.12allowsustocheckthat⏸fxy.SUzp;y[ٟ^3+zg⏲isanirredundantsystemofgeneratorsofI."NowLwewanttocomputethecolonidealLI]:lP M)(f),wheref`isthepGolynomial|f9=xMy[ٟ^2L.|ThelemmatellusthatwehavetocalculateSyz#1(xoy[ٟ^2L;UPxyHzp;y[ٟ^3zp).TheresultisthemoGdulegeneratedbythetwovec-tors@(y[;1;1)and(z txy[;xy[ٟ^2L;0).@Thusweobtain@I:lP Ʋ(f)=(y[;z txy)=(y[;zp)=p1|s."The-explanationofthisresultissimple,andwecanproveitdirectly*.For궵g"2p1|s,O]wehaveO]fg"2p1|s,andalsofg"2p2|s,sincefڧ2p2|s.Thereforewehave궵fgӸ2QI,BDandthismeansthatBDg2QIܲ:lP (f).Conversely*,BDletBDfgӸ2QI.Then궵fg"2p1ȲandUUf=ڧ2 Lkp1impliesg"2p1|s,UUbGecausep1isUUaprimeideal.PropQositionT3.2.15.w(ComputationTofColonIdeals)L}'etIM3=hg1|s;:::;gsF:iandN=hh1|s;:::;htVib}'eItwoPc-submodulesIofPc^r&,andletHβ=(h1|s;:::;htV).~oa)%The7c}'olonideal7N':lP OxMRandtheannihilatorofM=(N:B\#'M)c}'anbe%c}'omputedusingL}'emma3.2.13andtheformulauXYUN3:lP ƵM=AnncP|(M=(NO\8M))=^#s \ ti=1(N3:lPhgiTLi)LtHb)%Considerthefollowingblo}'ckmatrixofsizerGs8(st+1)1Wō>;1684`3.FirstTApplications-ō:~߸M=T0BBBfi@ 1g1 H730Jb0 1g2"qt05nHJC.Nß.SC.c:.c:.c:.I.I.I.#h.#h.#h.3.8+c.;3.2Elemen9taryTOpAerationsonModules9169-ō>;3.2.C ColonTMoQdules5궲GiventwoidealsIJandJw۲inaringRDz,wehaveseenhowtocomputethecolon idealIPI:lRJ9.IPThecolonidealofoneRDz-moGdulebyanotherwasageneralizationof thiscolonidealopGerationbetween ideals.Butthereisanotherway togeneralizeUUitfromidealstomoGdules,namelythecolonmoduleoperation.`CDe nitionT3.2.17.o3|LetaR(bGeaaring,letICbGeanidealinRDz,letU|bGean궵RDz-moGdule,andletMandNbGetwoRDz-submodulesofU.Thentheset궵N!:lM &I=fm2MjITPrmNgxisxanRDz-submoGduleofM̲(andofU).ItiscalledUUthecolonTmoQduleofUUNlpbyI7inM."ThePpurpGoseofthissubsectionistoexplainseveralmethodsforcomputingcolon8moGdulesof nitelygeneratedmodulesover8ane8K-algebras.W*e rstreduce*theproblemtothecaseofsubmoGdulesofa nitelygeneratedfree궵Pc-moGdule,UUwhereUUP*=K[x1|s;:::;xnq~]isapGolynomialringasabove.PropQositionT3.2.18.wL}'et촵Jbeanide}'alinP,letUb}'ea nitelygeneratedmo}'duleFovertheFK-algebraPV=J9,FandletFM]6andNb}'eFtwoPV=J9-submodulesofԗU.ԗF;urthermor}'e,letIyb}'eanidealinԗP8&containingJ9.ԗOurgoalistoc}'omputeN3:lM 8(I=J9)."Supp}'ose1wearegivenapresentation1UT͍Pڸ+3Pڲ=QPc^r1=V withaPc-submo}'duleVofPc^r1.Wec}'anwriteMT͍N+3N=M^0T=V>andNT͍N+3N=N^0T=V>withPc-submo}'dulesM^0and̔N^0ofPc^r c}'ontainingV8.̔ThenND:lM $(I=J9)isther}'esidueclassmoduleofN^0l:V³M0 rI\inU.Pr}'oof.6ThemoGduleƵN3:lM 8(I=J9)isgivenbyƸfa{vո2M^0T=Vj(I=J9)ø>v N^0=V8g.This8setis8fa{vո2M^0T=VjIv"N^0lforUUevery/v2M^0lwithUUresidueclassVwVOv^KѸ2Ug.Therefore=itistheimageof=fv"2M^0ljIcvN^0Tg=inU.=ThelastsetisnothingbutUUN^0l:V³M0 rI7whichUUprovestheclaim.zExampleT3.2.19.h`Let:RbGe:theK-algebrawithK-basisf1;"g,wherewehaveg"^28=50,gletNP=h(1;")iRǟ^2:,gandletg}fe.8IbGetheidealg}fe.8I Բ=(")ginRDz.SuppGoseUUwewanttocomputeUUN3:ȈR2 ֮}fe.8I."T*oۊthisend,we rstwriteۊRQintheformR=ŵPV=(x^2|s)withP T=K[x].Here "is theimageof xinRDz, and}fe.8I xistheimageoftheidealI=(x).ThenwenoticethatߵRǟ^27=RPc^2=VòforV6=h(x^2|s;0);(0;x^2)i,andthatߵNm=RN^0T=V궲for9QN^0l=h(1;x);(x^2|s;0);(0;x^2)i.9QThusthedesiredcolonmoGduleistheimageofUUN^0l:ȈP:2 I7inRǟ^2:."AsnaconsequenceofthispropGosition,weshallnowrestrictourattentiontoc#thecasewherec#Mz>andNarePc-submoGdulesc#ofPc^r 4TandI,isanidealinJPc.JInamannersimilartothelastsubsection,webGeginbyexplainingthe+kcomputationofthecolonmoGduleofasubmodulebyaprincipalideal.W*ecpresenttwomethoGdsfordoingthiscalculation,onebasedonasyzygymoGdulecomputation,andonebasedonperformingacertainintersectionoftwoUUsubmoGdules.ɮWō>;1704`3.FirstTApplications-ō>;LemmaT3.2.20.bG L}'etM7=hg1|s;:::;gsF:iandN=hh1|s;:::;htVib}'etwoPc-sub- mo}'dules!of!Pc^r&,andletf[2HP nl|f0g.F;urthermor}'e,letfv1|s;:::;vu:bgHPc^r Rb}'easystemofgener}'atorsofthePc-modulefMH\1N.F;ori=1;:::;u,wemaywritevid=fwi3forsomewi2M.Thenwehave썒N3:lM 8(f)=hw1|s;:::;wu:bi"Inp}'articular,wecomputeNt:lM T(f)asfollows.Letfa{~v1 UW;:::; #~v` \rg]Pc^s+t b}'easystemofgeneratorsofthemoduleSyzM%(fg1|s;:::;fgsF:;h1|s;:::;htV).Ifwewriteb~vjj=(f1j3;:::;fs+#t]jm)withf1j;:::;fs+#t]jJ2PvforjY=1;:::;`,thenx퍑rw$N3:lM 8(f)=h s 3 Pti=1fij gidj1jY`i'Pr}'oof.6Firstweobservethat˵fD0wi|=20vi2fMG\0NIKNimpliesthat궵wi2pNދ:lM (f)fori=1;:::;u.Conversely*,ifwestartwithanelement궵mr2N2:lM i7(f),wehavefmr2fMa\}FN.Thuswecanrepresentfmintheformgfm嫲=Pލ su% si=1ݵaiTLvi9=Pލ su% si=1aiTLfwiwithga1|s;:::;au 2Pc.gW*ecancelf{=andobtainUUm=Pލ USu% USi=1tJaiTLwid2hw1|s;:::;wu:bi,UUaswewantedtoshow."T*o provetheadditionalclaim,itsucestoapplyPropGosition3.2.3.atocomputeUUtheintersectionUUfMO\8N.މ썍ExampleT3.2.21.h`Whenweapplyparta)ofthislemmainthesituationof?thepreviousexample,weseethatwehavetocompute?xPc^2 _ʸ\ȵN^0 ݱ=궸h(x;0);(0;x)i\h(1;x);(x^2|s;0);(0;x^2)i.Theresultish(x;0);(0;x^2)i.There-fore wehave ߵN^0:ȈP:2uI=h(1;0);(0;x)i, andthecolonmoGduleN ²:ȈR2=}fe.8I>TwewereUUoriginallyinterestedinequalsUUh(1;0);(0;")i."F*orEthecomputationofcolonmoGdulesinthegeneralcase,wehaveagainthechoicebGetweenreductionstothecaseofprincipalidealsandadirectmethoGd.UUAswesawbGefore,itisenoughtotreatsubmodulesofUUPc^r1.PropQositionT3.2.22.w(ComputationTofColonMoQdules)L}'etٗM\q=EVhg1|s;:::;gsF:iandN=EVhh1|s;:::;htVib}'eٗtwoPc-submodulesٗofPc^r{,let궸G^=d(g1|s;:::;gsF:),letҸH}C=(h1;:::;htV),andletҵI-odPNab}'eanidealgeneratedbyasetofp}'olynomialsff1|s;:::;f`g.~oa)%Wemayc}'omputeN3:lM 8I\byusingLemma3.2.20andtheformula̍kN3:lM 8I=ʹ` \ ti=1-'N:lM(fiTL)e`Hb)%Considerthefollowingblo}'ckmatrixofsizerG`8(s+`t)+덑zbM=T0BBBfi@ 1f1|sG'WH=0QJib0 1f2|sG)m0<HQj<.U.Y<.jZ3.jZ3.jZ3.t.t.t.*.*.*.:.>\.Cܟ.Qj<.U.Y<.ib0 f`G)m0:TD0gdHTqҹ1qҹCqҹCqҹCfiqҹAݕWō>;3.2Elemen9taryTOpAerationsonModules9171-ō>;%L}'et{fv1|s;:::;vu:bgPc^s+`t0be{asystemofgeneratorsofthesyzygymodule ƍ%Syz4(M).*4Wewrite*4vj =_(f1j3;:::;fsj;f:(1)Z1j ;:::f:(1)Ztj ;::::::UF;f:(`)Z1j ,w;:::;f:(`)Ztj ,w)ˍ%withf1j3;:::;f:(`)Ztj 2PvforjY=1;:::;u.ThenwehavezjPN3:lM 8I=h s 3 Pti=1fij gidjjY=1;:::;uiLtPr}'oof.6PartXa)followsdirectlyfromthede nitions.Thereforeweproveclaimyb).Letyw1|s;:::;wsF:;w:D(1)l1 X;:::;w:D(1)͍t X;::::::UF;w:D(`)l1 ]˵;:::;w:D(`)͍tUDbGeythecolumnlvectorsEofEM.W*ehaveEPލ"s%"i=1fij wiV=X Pލ 8`% 8m=1(Pލ ;t% ;i=12f:(m)Zij,w:D(m)Zi)Eforevery w궵js=s1;:::;u.e%Ifweconsiderthee%kP^th ibatchofe%tcompGonentse%ofthisequation, cwe[seethat[fk됲(Pލ ;s% ;i=12fij giTL)г=Pލ ^t% ^i=1}f:(k+B)Zij ?!hidforjc>=1;:::;uandk!J=1;:::;`. 0HenceUUwegetUUPލ㐴s%i=1fij gid2N3:lM 8I7forUUjY=1;:::;u. "Conversely*,letõvO=0vPލs%i=1ݨaiTLgi¸20vNG:lM 'Iwitha1|s;:::;as v2Pc.Then U^thereexistpGolynomialsa11x;:::;at` i2-Psuchthatfk됵v1=-Pލcht%chi=1_aikܵhiIfor궵k=1;:::;`.UUBycombiningtheseequationsintoavectorequation,wegetc궵a1|sw1S+ g+asF:ws(a11xw:D(1)l1 8++at1ɵw:D(1)͍t 8+ /=+a1`Yw:D(`)l1 ++at`