Robbiano, Kreuzer - Computational Commutative Algebra.pdf - Algebra - besia86

MartinKreuzerandLorenzoRobbiano

Computational

Commutative

Algebra1

July3,2000

Springer-Verlag

BerlinHeidelbergNewYork

LondonParisTokyo

HongKongBarcelona

Budapest

Foreword

Hofstadter’sLaw:Italwaystakeslongerthanyouthinkitwilltake,

evenifyoutakeintoaccountHofstadter’sLaw.

(DouglasR.Hofstadter)

DearReader,

whatyouareholdinginyourhandsnowisforyou abook .Butforus,forour

familiesandfriends,ithasbeenknownas thebook overthelastthreeyears.

Threeyearsofintenseworkjusttoﬁllthreecentimetersofyourbookshelf!

Thisamountstoaboutonecentimeterperyear,orroughlytwo-ﬁfthsofan

inchperyearifyouarenon-metric.Clearlywehadampleopportunityto

experiencethefullforceofHofstadter’sLaw.

WritingabookaboutComputationalCommutativeAlgebraisnotun-

likecomputingaGr¨obnerbasis:youneedunshakeablefaithtobelievethat

theprojectwilleverend;likewise,youmusttrustintheNoetherianityof

polynomialringstobelievethatBuchberger’sAlgorithmwilleverterminate.

Naturally,wehopethattheﬁnalresultprovesoure®ortsworthwhile.This

isabookforlearning,teaching,reading,and,mostofall,enjoyingthetopic

athand.

SinceneitherofusisanativeEnglishspeaker,theliteraryqualityof

thisworkisnecessarilyalittlelimited.Worriesaboutourlackoflinguis-

ticsophisticationgrewconsiderablyuponreadingthefollowingpartofthe

introductionof“TheRandomHouseCollegeDictionary”

Aneducatedspeakerwilltransferfrominformal haven’t toformal have

not .Theuneducatedspeakerwhoinformallyuses Iseen or Idonegone

mayadjusttotheformalmodewith Ihavesaw and Ihavewent .

Quiteapartfrombeingunabletodistinguishbetweentheinformaland

formalmodes,wewerefrequentlypuzzledbysuchelementaryquestionsas:

isthereanotherwordforsynonym?Luckily,wewereabletoextricateour-

selvesfromtheworstmiresthankstothegenerousaidofJohnAbbottand

TonyGeramita.TheyprovideduswithmuchinsightintoBritishEnglishand

AmericanEnglish,respectively.However,notwithstandingtheirilluminating

help,weweresometimesunabletodiscovertheultimatetruth:should I be

anideal in aring R oranideal of aring R ?Finally,wedecidedtobe

non-partisananduseboth.

Havingrevealedthenamesoftwoofourmainaides,wenowabandonall

pretenceandadmitthat thebook isreallyajointe®ortofmanypeople.We

especiallythankAlessioDelPadronewhocarefullycheckedeverydetailof

themaintextandtest-solvedalloftheexercises.Thetasksofproof-reading

andcheckingtutorialswerevariouslycarriedoutbyJohnAbbott,AnnaBi-

gatti,MassimoCaboara,RobertForkel,TonyGeramita,BettinaKreuzer,

andMarieVitulli.AnnaBigattiwroteorimprovedmanyoftheCoCoApro-

gramswepresent,andalsosuggestedthetutorialsaboutToricIdealsand

DiophantineSystemsandIntegerProgramming.ThetutorialaboutStrange

PolynomialscomesfromresearchbyJohnAbbott.ThetutorialaboutElim-

inationofModuleComponentscomesfromresearchinthedoctoralthesisof

MassimoCaboara.ThetutorialaboutSplineswasconceivedbyJensSchmid-

bauer.Mosttutorialsweretested,andinmanycasescorrected,bythestu-

dentswhoattendedourlecturecourses.OurcolleaguesBrunoBuchberger,

DavePerkinson,andMossSweedlerhelpeduswithmaterialforjokesand

quotes.

Moralhelpcamefromourfamilies.OurwivesBettinaandGabriella,and

ourchildrenChiara,Francesco,Katharina,andVeronikapatientlyhelpedus

toshouldertheproblemsandburdenswhichwritingabookentails.Andfrom

thepracticalpointofview,thisprojectcouldneverhavecometoasuccessful

conclusionwithouttheuntiringsupportofDr.MartinPeters,hisassistant

RuthAllewelt,andtheothermembersofthesta®atSpringerVerlag.

Finally,wewouldliketomentionourfavouritesoccerteams,Bayern

M¨unchenandJuventusTurin,aswellasthestockmarketmaniaofthelate

1990s:theyprovideduswithnever-endingmaterialfordiscussionswhenour

workon thebook becametoooverwhelming.

MartinKreuzerandLorenzoRobbiano,

RegensburgandGenova,June2000

Contents

Foreword ..................................................... V

Introduction .................................................. 1

0.1WhatIsThisBookAbout?.............................. 1

0.2WhatIsaGr¨obnerBasis?............................... 2

0.3WhoInventedThisTheory?............................. 3

0.4Now,WhatIsThisBook Really About?................... 4

0.5WhatIsThisBook Not About?.......................... 7

0.6AreThereanyApplicationsofThisTheory?............... 8

0.7HowWasThisBookWritten?............................10

0.8WhatIsaTutorial?.....................................11

0.9WhatIsCoCoA?........................................12

0.10AndWhatIsThisBookGoodfor?.......................12

0.11SomeFinalWordsofWisdom............................13

1.Foundations .............................................. 15

1.1PolynomialRings.......................................17

Tutorial1. PolynomialRepresentationI ....................24

Tutorial2. TheExtendedEuclideanAlgorithm ..............26

Tutorial3. FiniteFields .................................27

1.2UniqueFactorization....................................29

Tutorial4. EuclideanDomains ...........................35

Tutorial5. SquarefreePartsofPolynomials .................37

Tutorial6. Berlekamp’sAlgorithm ........................38

1.3MonomialIdealsandMonomialModules..................41

Tutorial7. Cogenerators ................................47

Tutorial8. BasicOperationswithMonomialIdealsandModules 48

1.4TermOrderings........................................49

Tutorial9. MonoidOrderingsRepresentedbyMatrices ........57

Tutorial10. ClassiﬁcationofTermOrderings ................58

1.5LeadingTerms.........................................59

Tutorial11. PolynomialRepresentationII ..................65

Tutorial12. SymmetricPolynomials .......................66

Tutorial13. NewtonPolytopes ...........................67

VIII Contents

1.6TheDivisionAlgorithm.................................69

Tutorial14. ImplementationoftheDivisionAlgorithm ........73

Tutorial15. NormalRemainders ..........................75

1.7Gradings..............................................76

Tutorial16. HomogeneousPolynomials ....................83

2.Gr¨obnerBases ............................................ 85

2.1SpecialGeneration.....................................87

Tutorial17. MinimalPolynomialsofAlgebraicNumbers ......89

2.2RewriteRules..........................................91

Tutorial18. AlgebraicNumbers ..........................97

2.3Syzygies...............................................99

Tutorial19. SyzygiesofElementsofMonomialModules .......108

Tutorial20. LiftingofSyzygies ...........................108

2.4Gr¨obnerBasesofIdealsandModules.....................110

2.4.AExistenceofGr¨obnerBases.......................111

2.4.BNormalForms...................................113

2.4.CReducedGr¨obnerBases...........................115

Tutorial21. LinearAlgebra ..............................119

Tutorial22. ReducedGr¨obnerBases .......................119

2.5Buchberger’sAlgorithm.................................121

Tutorial23. Buchberger’sCriterion ........................127

Tutorial24. ComputingSomeGr¨obnerBases ...............129

Tutorial25. SomeOptimizationsofBuchberger’sAlgorithm ...130

2.6Hilbert’sNullstellensatz.................................133

2.6.ATheField-TheoreticVersion.......................134

2.6.BTheGeometricVersion...........................137

Tutorial26. GraphColourings ...........................143

Tutorial27. A±neVarieties .............................143

3.FirstApplications ........................................ 145

3.1ComputationofSyzygyModules.........................148

Tutorial28. Splines ....................................155

Tutorial29. Hilbert’sSyzygyTheorem .....................159

3.2ElementaryOperationsonModules.......................160

3.2.AIntersections....................................162

3.2.BColonIdealsandAnnihilators.....................166

3.2.CColonModules..................................169

Tutorial30. ComputationofIntersections ..................174

Tutorial31. ComputationofColonIdealsandColonModules ..175

3.3HomomorphismsofModules.............................177

3.3.AKernels,Images,andLiftingsofLinearMaps........178

3.3.BHom-Modules...................................181

Tutorial32. ComputingKernelsandPullbacks ..............191

Tutorial33. TheDepthofaModule .......................192

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