理想、簇與算法 第三版 英文

Preface to the First Edilion
Preface to the Second Edition
Preface to the Third Edition

1.Geometry,Algebra,and Algorithms
§1.Polynomials andAffine Space
§2.Affine Varieties
§3.Parametfizafions of Affine Varieties
§4.Ideals
§5.Polynomials of One Variable

2.GroebnerBases
§1.Introduction
§2.Orderings on th eMonomials in k[x1，…xn]
§3.A Division Algorithm in k[x1,…，xn]
§4.Monomial Ideals and Dickson’’S Lemma
§5.The Hilbert Bails Theorem and Groebner Bases
§6 Properties of Groebner Bases
§7.Buchberger’’S Algorithm
§8.First Applicafions of Groebner Bases
§9.(Optional)Improvements On Buchberger’’S Algorithm

3.EliminaUon Theory
§1.The Eliminationand Extension Theorems
§2.The Geometry of Elimination
§3.Implicitization
§4.Singular Pointsand Envelopes
§5.Unique Factorization and Resultants
§6.Resultants and the Extension Theorem

4.The Algebra-Geometry Dictionary
§1.Hilbert’’s Nullstellensatz
§2.Radical Ideals and the Ideal-Variety Correspondence
§3.Sums，Prodects，and Intersections of ldeals
§4.Zariski Closure and Quotiens of Ideals
§5.Irreducible Varieties and Prime Ideals
§6.Decomposition of a Varietyinto Irreducibles
§7.(Optional)Primary Decomposition of Ideals
§8.Summary

5.Polyaomial and Rational Functionson a Vanety
§1.Polynomial Mappings
§2.Quotients of Polynomial Rings
§3.Algorithmic Compumtionsin k[x1，…,xn]/I
§4.The Coordinate Ring of an Affine Variety
§5.Rational Functions on a Variety
§6.(Optional)Proof of the Closure Theorem

6.Robotics and Automatic Geometric Theorem Proving
§1.Geometric Description of Robots
§2.The Forward Kinematic Problem
§3.The Inverse Kinematic Problem and Motion Planning
§4.Automatic Geometric Theorem Proving
§5.Wu’’s Method

7.Invariant Theory of Finite Groups
§1.Symmetric Polynomials
§2.Finite Matrix Groups and Rings of lnvariants
§3.Generators for the Ring of Invariants
§4.Relations Among Generators and the Geometry of Orbits

8.Projective Algebraic Geometry
§1.TheProjective Plane
§2.Projective Space andProjectiveVarieties
§3.the Projective Algebra-Geometry Dictionary
§4.The Projective Closure of an AffineVariety
§5.Projective Elimination Theory
§6.The Geometry of Quadric Hypersurfaces
§7.Bezout’’STheorem

9.The Dimension of a Variety
§l.The Variety ofa Monomial Ideal
§2.The Complement of a Monomial Ideal
§3.The Hilbcrt Function and the Dimension of a Variety
§4.Elementary Properties of Dimension
§5.Dimension and Algebraic Independence
§6.Dimension and Nonsingularity
§7.The Tangent Cone

Appendix A.Some Concepts from Algebra
§1.Fields andRings
§2.Groups
§3.Determinants
Appendix B.Pseudocode
§1.Inputs，Outputs,Variables，and Constants
§2.Assignment Statements
§3.Looping Structures
§4.Branching Structures
Appendix C.Computer Algebra Systems
§1.AXIOM.
§3.Mathematica
§4.REDUCE
§5.Other Systems
Appendix D Independent Projects
§1.General Comments
§2.Suggested Projects
References
Indes