算術(shù)代數(shù)：英文本

Preface
1Introduction
1.1Prologue:AlgebraandAlgorithms
1.2Motivations
1.2.1ConstructiveAlgebra
1.2.2AlgorithmicandComputationalAlgebra
1.2.3SymbolicComputation
1.2.4Applications
1.3AlgorithmicNotations
1.3.1DataStructures
1.3.2ControlStructures
1.4Epilogue
BibliographicNotes
2AlgebraicPreliminaries
2.1IntroductiontoRingsandIdeals
2.1.1RingsandIdeals
2.1.2Homomorphism,ContractionandExtension
2.1.3IdealOperations
2.2PolynomialRings
2.2.1Dickson'sLemma
2.2.2AdmissibleOrderingsonPowerProducts
2.3GrSbnerBases
2.3.1GrSbnerBasesinK[x1,x2,...,xn]
2.3.2Hilbert'sBasisTheorem
2.3.3FiniteGrobnerBases
2.4ModulesandSyzygies
2.5S-Polynomials
Problems
SolutionstoSelectedProblems
BibliographicNotes
3ComputationalIdealTheory
3.1Introduction
3.2StronglyComputableRing
3.2.1Example:ComputableField
3.2.2Example:RingofIntegers
3.3HeadReductionsandGrSbnerBases
3.3.1AlgorithmtoComputeHeadReduction
3.3.2AlgorithmtoComputeGrSbnerBases
3.4DetachabilityComputation
3.4.1ExpressingwiththeGrSbnerBasis
3.4.2Detachability
3.5SyzygyComputation
3.5.1SyzygyofaGrSbnerBasis:SpecialCase
3.5.2SyzygyofaSet:GeneralCase
3.6Hilbert'sBasisTheorem:Revisited
3.7ApplicationsofGrSbnerBasesAlgorithms
3.7.1Membership
3.7.2Congruence,SubidealandIdealEquality
3.7.3SumandProduct
3.7.4Intersection
3.7.5Quotient
Problems
SolutionstoSelectedProblems
BibliographicNotes
4SolvingSystemsofPolynomialEquations
4.1Introduction
4.2TriangularSet
4.3SomeAlgebraicGeometry
4.3.1DimensionofanIdeal
4.3.2Solvability:Hilbert'sNullstellensatz
4.3.3FiniteSolvability
4.4FindingtheZeros
Problems
SolutionstoSelectedProblems
BibliographicNotes
5CharacteristicSets
5.1Introduction
5.2PseudodivisionandSuccessivePseudodivision
5.3CharacteristicSets
5.4PropertiesofCharacteristicSets
5.5Wu-RittProcess
5.6Computation
5.7GeometricTheoremProving
Problems
SolutionstoSelectedProblems
BibliographicNotes
6AnAlgebraicInterlude
6.1Introduction
6.2UniqueFactorizationDomain
6.3PrincipalIdealDomain
6.4EuclideanDomain
6.5GaussLemma
6.6StronglyComputableEuclideanDomains
Problems
SolutionstoSelectedProblems
BibliographicNotes
7ResultantsandSubresultants
7.1Introduction
7.2Resultants
7.3HomomorphismsandResultants
7.3.1EvaluationHomomorphism
7.4RepeatedFactorsinPolynomialsandDiscriminants
7.5DeterminantPolynomial
7.5.1Pseudodivision:Revisited
7.5.2HomomorphismandPseudoremainder
7.6PolynomialRemainderSequences
7.7Subresultants
7.7.1SubresultantsandCommonDivisors
7.8HomomorphismsandSubresultants
7.9SubresultantChain
7.10SubresultantChainTheorem
7.10.1Habicht'sTheorem
7.10.2EvaluationHomomorphisms.
7.10.3SubresultantChainTheorem
Problems
SolutionstoSelectedProblems
BibliographicNotes
8RealAlgebra
8.1Introduction
8.2RealClosedFields
8.3BoundsontheRoots
8.4Sturm'sTheorem
8.5RealAlgebraicNumbers
8.5.1RealAlgebraicNumberField
8.5.2RootSeparation,Thom'sLemmaandRepresentation
8.6RealGeometry
8.6.1RealAlgebraicSets
8.6.2Delineability
8.6.3Tarski-SeidenbergTheorem
8.6.4RepresentationandDecompositionofSemialgebraicSets
8.6.5CylindricalAlgebraicDecomposition
8.6.6TarskiGeometry
Problems
SolutionstoSelectedProblems
BibliographicNotes
AppendixA:MatrixAlgebra
A.1Matrices
A.2Determinant
A.3LinearEquations
Bibliography
Index