本书是中国科学院**的研究生原版教材之一，是近年来出版的计算机代数方面的权威著作．书中全面介绍了近20年来该领域的主要成果，包括Grobner基、Wu-Ritt特征基、系统式法、实代数几何等。这些成果是计算机与代数几何交叉研究所产生的新成果，不仅对代数的发展有很大的影响，也对代数学算法在机器人、计算机视觉等方面的应用提供了基础。本书可作为数学系及计算机系相关专业研究生的教材。

Preface
I Introduction
l.l Prologue: Algebra and Algorithms
l.2 Motivations
l.2.l Constructive Algebra
l.2.2 Algorithmic and Computational Algebra
l.2.3 Symbolic Computation
l.2.4 Applications
l.3 Algorithmic Notations
l.3.l Data Structures
l.3.2 Control Stractures
l.4 Epilogue Bibliographic Notes
2 Algebralc Preliminaries
2.l Introduction to Ring and Ideals
2.l.l Rings and Ideak
2.l.2 Homomorphism Contraction and Extension
2.l.3 Ideal Operationn
2.2 Polynomial Rings
2.2.l Dickson's Lemma
2.2.2 Admisaible Ordering on Power Products
2.3 Gr8bner Bases
2.3.2 Hilbert's Basis Theorem
2.3.3 Finite Grtibner Bases
2.4 Modules and Syzygies
2.5 s-Polynomiak
Problems
Solutiona to Selected Problems
Bibliographic Notes
3 Computational Ideal Theory
3.l Introduction
3.2 Strongly Computable Ring
3.2.l Example: Computable Field
3.2.2 Example: Ring of Integere
3.3 Head
3.3.1 Algorithm to Gompute Head
3.3.2 Algorithm to Compute Gr6bner
3.4 Detachability Computation
3.4.1 Expresing with the Gr6bner Basis
3.4.2 Detacltability
3.5 Syzygy Computation
3.5.l Syzgy of a Gr6bner Basis: Special Caae
3.5.2 Syzgg of a Set: General Caae
3.6 Hilbertases'e Basis Theorem: Revisited
3.7 Applications of Gr8bner Bases Algorithms
3.7.l Membership
3.7.2 Congruence, Subideal and Ideal Equality
3.7.3 Sum and Product
3.7.4 Intersection
3.7.5 Quotient
Problems
Solutione to Selected Problems
Bibliographic Notes
4 Solving Syetems of Polynomial Equationn
4.l Introductioll
4.2 Triangular Set
4.3 Some Algebraic Geometry
4.3.l Dimenaion of an Ideal
4.3.2 Solvability : Hilbert's Nulktellensatz
4.3.3 Finite Solv&bility
4.4 Finding the Zeros
Problems
Solutions to Selected Problems
Bibliographic Notes
5 Charac*eristic Sete
5.l Introductioll
5.2 Pseudodivision and Successive Peeudodivision
5.3 Characteristic Sets
5.4 Properties of Characteristic Sets
5.5 Wu-Ritt Process
5.6 Computation
5.7 Geometric Theorem Proving
Problems
Solutions to Selected Problema
Bibliographic Notes
6.l Introduction
6.2 Unique Factorization Domain
6.3 Principal Ideal Domain
6.4 Euclidean Domain
6.5 Gauss lemma
6.6 Strongly Computable Euclidean Domains
Problems
Solutions to Selected Problems
Bibliographic Notes
7 Resultants and Subresultants
7.l Introduction
7.2 Reultants
7.3 Homomorphisme alld Resultants
7.3.l Evaluation Homomorphism
7.4 Repeated Factors in Polynomials tiad Discriminants
7.5 Determinant Polynomial
7.5.l Pseudodivision : Rsvisit4sid
7.5.2 Homomorphism and Pseudoremalllder
7.6 Polynomial Remainder Sequences
7.7 Subresultants
7.7.l Subresult&nts and Common Divieors
7.8 Homomorphisms alld Subresultants
7.9 Subresultant Chain
7.1O Subresultant Chain Theorem
7.1O.l Habicht Ig Theorem
7.1O.2 Evaluation Homomorphinme
7.1O.3 Subresultant Chain Theorem
Problems
Solutiona to Selected Problems
Bibliographic Notes
8 Real Algebra
8.l Introduction
8.2 Real Closed Fields
8.3 Bounds on the Roots
8.4 Sturm's Theorenl
8.5 Real Algebraic Numbers
8.5.l Real Algebraic Numbers
8.5.2 Root Separation, Thom'e lemma and Representat
8.6 Real Geometry
8.6.l Real Algebraic Sets
8.6.2 Delineability
8.6.3 Tarski-Seidenberg Theorem
8.6.4 Rspruentation and Decomposition of Semialgebraic sets
8.6.5 Cylindrical Algebraic Decompoeition
8.6.6 Tarski Geometry
Problems
Solutions to Selected Problema
Bibliographic Notes
Appendix A: Matrix Algebra
A.l R4atrica
A.2 Detemlillabnt
A.3 Linear Equations
Bibliography
Index