考克斯著的《理想簇与算法(第3版)》是Springer出版的《大学生数学教材》丛书之一，1992年首次出版，1997年第2版，这是第三版，将原来的版本作了若干修订，是一本交换代数与计算代数几何的教科书，各章均附有相当数量的练习题。阅读本书需要的预备知识为抽象代数基础。本书可作为大学有关专业的教材或教学参考书，亦可供有关方面的研究生和研究人员参考阅读。目次：几何、代数与算法；Groebner基；消去理论；代数几何辞典；簇上的多项式与有理函数；自动机与几何定理的机械化证明；有限群的不变量理论；射影代数几何；簇的维数。

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