理想数、簇与算法 第2版 作者：Cox，D.等著 出版时间：2004年版 内容简介 We wrote this book to introduce undergraduates to some interesting ideas in algebraic geometry and commutative algebra。 Until recently，these topics involved a lot of abstract mathematics and were only taught in graduate school。 But in the 1960s，Buchberger and Hironaka discovered new algorithms for manipulating systems of polynomial equations。 Fueled by the development of computers fast enough to run these algorithms，the last two decades have seen a minor revolution in commutative algebra。 The ability to compute efficiently with polynomial equations has made it possible to investigate complicated examples that would be impossible to do by hand，and has changed the practice of much research in algebraic geometry。 This has also enhanced the importance of the subject for computer scientists and engineers，who have begun to use these techniques in a whole range of problems。 目录 PrefacetotheFirstEdition PrefacetotheSecondEdition 1. Geometry,cAlgebra,candAlgorithms 1. PolynomialsandAffineSpace 2. AffineVarieties 3. ParametrizationsofAffineVarieties 4. Ideals 5. PolynomialsofOneVariable 2. GroebnerBases 1. Introduction 2. OrderingscontheMonomialsink[x1. ,....,xn] 3. ADivisionAlgorithmink[x1. ,....,xn] 4. MonomialIdealsandDickson'scLemma 5. TheHilbertBasisTheoremandGroebnerBases 6. .PropertiesofGroebnerBases 7..Buchberger'scAlgorithm 8. .FirstApplicationsofGroebnerBases 9.(Optional)ImprovementsonBuchberger'scAlgorithm 3. EliminationTheory 1. TheEliminationandExtensionTheorems 2. TheGeometryofElimination 3. Implicitization 4. SingularPointsandEnvelopes 5. UniqueFactorizationandResultants 6. ResultantsandtheExtensionTheorem 4. TheAlgebra-GeometryDictionary 1. Hilbert'sNullstellensatz 2. RadicalIdealscandtheIdeal-VarietyCorrespondence 3. Sums,cProducts,candIntersetionscofIdeals 4. ZariskiClosureandQuotientscofIdeals 5. IrreducibleVarietiesandPrimeIdeals 6. DecompositionofaVarietycintoIrreducibles 7.(Optional)PrimaryDecompositionofIdeals 8. Summary 5. PolynomialandRationalFunctionsonaVariety 1. PolynomialMappings 2. QuotientsofPolynomialRings 3. AlgorithmicComputationscink[x1. ,....,xn]I 4. TheCoordinateRingofanAffineVariety 5. RationalFunctionsconcaVariety 6. (Optional)ProofcoftheClosureTheorem 6. RoboticsandAutomaticGeometricTheoremProving 1. GeometricDescriptionofRobots 2. TheForwardKinematicProblem 3. TheInverseKinematicProblemandMotionPlanning 4. AutomaticGeometricTheoremProving 5. Wu'sMetho 7.InvariantTheoryofFiniteGroups 1. SymmetricPolynomials 2. FiniteMatrixGroupsandRingsofInvariants 3. GeneratorsfortheRingofInvariants 4. RelationsAmongGeneratorsandtheGeometryofOrbits 8. ProjectiveAlgebraicGeometry 1. TheProjetivePlane 2. ProjectiveSpaceandProjectiveVarieties 3. TheProjectiveAlgebra-GeometryDictionary 4. TheProjectiveClosureofanAffineVariety 5. ProjectiveEliminationTheory 6. TheGeometryofQuadricHypersuffaces 7. Bezout'sTheorem 9.TheDimensionofaVariety 1. TheVarietyofaMonomialIdea 2. heComplementofaMonomialIdeal 3. TheHilbertFunctionandtheDimensionofaVariety 4. ElementarycPropertiescofcDimension 5. DimensionandAlgebraicIndependence 6. DimensionandNonsingularity 7. TheTangentCone AppendixA.SomeConceptscfromAlgebra 1. FieldsandRings 2. Groups 3. Determinants AppendixB.Pseudocode 1. Inputs,Outputs,Variables,andConstants 2. AssignmentStatements 3. LoopingStructures 4. BranchingStructures AppendixC.ComputerAlgebraSystems 1. AXIOM 2. Maple 3. Mathematica 4. REDUCE 5. OthercSystems AppendixcD.cIndependentcProjects 1. GeneralcComments 2. SuggestedcProjects References Index
