Solving Polynomial Equation Systems IV: Volume 4, Buchberger Theory and Beyond

834 Pages·2016·3.744 MB·English

by  Teo Mora

SOLVING POLYNOMIAL EQUATION SYSTEMS
VolumeIV:BuchbergerTheoryandBeyond
InthisfourthandfinalvolumetheauthorextendsBuchberger’salgorithminthree
differentdirections.First,heextendsthetheorytogroupringsandotherOre-like
extensions,andprovidesanoperativeschemethatallowsonetosetaBuchberger
theoryoveranyeffectiveunitaryring.Second,hecoverssimilarextensionsastools
fordiscussingparametricpolynomialsystems,thenotionofSAGBIbases,Gröbner
basesoverinvariantringsandHironaka’stheory.Finally,MorashowshowHilbert’s
followers–notablyJanet,GuntherandMacaulay–anticipatedBuchberger’sideas
anddiscussesthemostpromisingrecentalternativesbyGerdt(involutivebases)and
Faugère(F andF ).
4 5
Thiscomprehensivetreatmentinfourvolumesisasignificantcontributionto
algorithmiccommutativealgebrathatwillbeessentialreadingforalgebraistsand
algebraicgeometers.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
EncyclopediaofMathematicsandItsApplications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityforDistributed
ParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopological
GraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,Computer
Science,andEngineering
135 V.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,
Geometry,andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethod
forMaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarcíaandÁ.RodríguezPalaciosNon-AssociativeNormedAlgebrasI:The
Vidav–PalmerandGelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond
159 V.BerthéandM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonic
Analysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractional
Problems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
Encyclopedia of Mathematics and Its Applications
Solving Polynomial Equation Systems
Volume IV: Buchberger Theory and Beyond
TEO MORA
UniversityofGenoa
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ISBN–VolumeI 978-0-521-81154-5Hardback
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Contents
Adieu pageix
PARTSEVEN: Beyond 1
46 Zacharias 3
46.1 Buchberger–ZachariasTheory 5
46.2 Kandri-Rody–Kapur: Strong Gröbner Bases over
aEuclideanDomain 27
46.3 Pan:StrongGröbnerBasesoveraPrincipalIdealDomain 31
46.4 ZachariasRings 37
46.5 Möller:GröbnerBasesoverZachariasRings 42
46.6 Möller:GröbnerBasesoveraPrincipalIdealRing 48
46.7 Useless S-pairs and Gebauer–Möller Set in Principal
IdealRings 50
46.8 Möller: Structural Theorem for Gröbner Bases over
aPrincipalIdealRing 54
46.9 Lazard’sStructuralTheoremoveraPrincipalIdealRing 56
46.10 OreConditions 63
46.11 OreTheory 68
46.12 Lazard’s Structural Theorem for Ore Extensions
overaPrincipalIdealDomain 83
46.13 MultivariateOreExtensions 85
46.14 IdealofVanishingPolynomial 96
47 Bergman 100
47.1 MöllerLeftLiftingTheoremforFreeAssociativeAlgebras 102
47.2 LeftGröbnerBasesforFreeAssociativeAlgebras 107
47.3 Möller’sRightLiftingTheoremandRightGröbnerBases
forFreeAssociativeAlgebras 115
47.4 BilateralGröbnerBases:TheLiftingTheorem 117
47.5 BilateralGröbnerBases:Pritchard’sModule 121
47.6 Bilateral Gröbner Bases of an Ideal in a Free Monoid
RingsoveraField 124
vi Contents
47.7 Pritchard’sProcedure 144
47.8 LaScala–Levandovskyy:LetterplaceIdeals 148
47.9 GröbnerBasisforWeakerAlgebras 157
47.10 Green:PathAlgebras 164
47.11 Reinert:BuchbergerTheoryonFunctionRings 172
47.12 deGraaf–Wisliceny–Gerritzen:GröbnerBases
onMagmaAlgebras 184
48 Ufnarovski 191
48.1 UfnarovskiGraph 192
48.2 Ufnarovski: Finite-State Automata and Infinite Gröbner
BasisComputation 198
48.3 2-VariableGraph 206
48.4 Shirayanagi:WordTree 207
48.5 Anick’sResolution 212
48.6 Borges:EliminationOrderingsandSymmetricGroups 217
48.7 TaxonomyofTermOrderings 221
48.8 IdealTheoreticalOperation 246
49 Weispfenning 256
49.1 Apel–Lassner:LieAlgebrasandBuchbergerTheory 258
49.2 Bergman:DiamondLemma 261
49.3 SkewandSolvablePolynomialRings 264
49.4 KredelConditions 268
49.5 Kandri-Rody–Weispfenning:SolvablePolynomialRings 271
49.6 TwistedPolynomialRings 278
49.7 Structural Theorem for Bilateral Ideals of a Twisted
PolynomialRing 284
49.8 Gateva’sMonoid 288
49.9 Gateva:NoetherianityofFinitelyPresentedAlgebras 291
49.10 Gateva:BinomialSkewPolynomialRings 298
49.11 WeispfenningRings 314
49.12 La Scala–Levandovskyy: Ore Extension of Multivariate
PolynomialRings 324
49.13 LaScala:BuchbergerTheoryforPartialDifferenceIdeals 332
50 Spear2 335
50.1 Zacharias’CanonicalRepresentation 336
50.2 TermOrdering 349
50.3 Pseudovaluations 351
50.4 GröbnerBasesandBuchbergerTheory 359
50.5 Möller’sLiftingTheorem 363
50.6 Spear’sTheorem 371
50.7 Pseudovaluations(2) 377
50.8 TowardBuchberger’sAlgorithm 381
50.9 Buchberger’sAlgorithms 383
50.10 TheArithmeticoftheAssociatedGradedRing 386
Contents vii
50.11 Noetherianity 394
50.12 CanonicalForms 397
50.13 CaseStudies 399
51 WeispfenningII 423
51.1 ComprehensiveGröbnerBases 425
51.2 Montes’Algorithm 429
51.3 CanonicalComprehensiveGröbnerBases 436
51.4 Wibmer:RegularFunctions 441
51.5 Kapuretal.:BranchingviaSolvingTechniques 444
51.6 Montes–Wibmer:CanonicalGröbnerCover 446
51.7 VonNeumannRegularRings 449
51.8 GröbnerBasesoverVonNeumannRegularRings 454
51.9 ComprehensivenessofGröbnerBasesoverVonNeumann
RegularRings(1) 457
51.10 Suzuki–Sato:AlternativeComprehensiveGröbnerBases 459
51.11 ComprehensivenessofGröbnerBasesoverVonNeumann
RegularRings(2) 465
52 Sweedler 468
52.1 ValuationRings 471
52.2 Mosteing–Sweedler:ValuationsandFiltrations 492
52.3 Mosteing–Sweedler:CharacterizingTermOrderFiltration 510
52.4 SubalgebraBases 514
52.5 SAGBIBasesforPolynomialInvariantsof
AlternatingGroups 519
52.6 FinitenessofSAGBIBases 528
52.7 Göbel’sBasisforInvariantRings 531
52.8 Buchberger’sAlgorithmforSymideals 536
52.9 Aschenbrenner–Hillar:BuchbergerTheory
forSymmetricIdeals 542
52.10 VariationsonSAGBI 552
53 Hironaka 565
53.1 ATheoryofNormalForms 567
53.2 NormalFormsandStandardBases 572
53.3 TheZariskificationofthePolynomialRing 576
53.4 TheTangentConeAlgorithm 581
53.5 StandardBasisAlgorithms 587
53.6 TheNormalFormAlgorithm 589
53.7 L-adicValuation,Localization,Zariskification 592
53.8 StandardBasesinLocalRings 593
54 HironakaII 597
54.1 Alonso–Luengo–RaimondoModelforAlgebraicPowerSeries 597
54.2 CanonicalFormsintheRingofAlgebraicPowerSeries 601
54.3 StandardBasesintheRingofAlgebraicPowerSeries 604
54.4 AnEffectiveVersionoftheWeierstrassPreparationTheorem 611
viii Contents
54.5 Alonso–Castro–Hauser:AnEffectiveVersionofHironaka’s
HenselianWeierstrassTheorem 620
54.6 BranchesofaCurveataSingularPoint 622
54.7 TowardMacMillan’sAlgorithm 626
54.8 MacMillan’sAlgorithm 630
55 Janet 633
55.1 JanetDecomposition 635
55.2 Riquier–JanetCompletion 651
55.3 Degree-compatibleTermOrderingandConvergency 658
55.4 JanetandHilbert 661
55.5 Janet:Involutiveness 664
55.6 Term Orderings and the Borel Property in Riquier
andHisFollowers 682
56 MacaulayV 691
56.1 Macaulay’sFormula 692
56.2 HilbertFunction 700
56.3 Sperner’sProofofMacaulay’sFormula 702
56.4 Gunther: Borel Relation and Borel Invariancy of
GenericForms 708
56.5 Gunther:Macaulay-likeFormulafortheBorelSet 716
56.6 Gunther’sProofofMacaulay’sFormula 720
56.7 Gunther:HilbertFunctionComputation 729
56.8 Gunther:Riquier–JanetProcedure 734
56.9 Bigatti–Hulett:BettiNumbersandLexSegments 740
56.10 Marinari–Ramella:RevlexSegments 744
56.11 Cioffi–Roggero:J-markedFamilies 750
57 GerdtandFaugère 757
57.1 Gerdt 758
57.2 Faugère 778
57.3 Mayr:Space-efficientGröbnerBasisComputation 796
57.4 Congedo: Buchberger–Wiesinger-Widi Groebnerian-
GaussianBound 800
Bibliography 803
Index 813