Some tapas of computer algebra.计算机代数一些Tapas

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作者: Arjeh M. Cohen著

出 版 社: 广东教育出版社

出版时间: 1998-12-1字数:版次:页数: 352印刷时间: 1998/12/01开本: 16开印次:纸张: 胶版纸I S B N : 9783540634805包装: 精装内容简介

This book arose from a series of courses on computer algebra which were given at Eindhoven Technical University. Its chapters present a variety of topics in computer algebra at an accessible (upper undergraduate/graduate) level with a view towards recent developments. For those wanting to acquaint themselves somewhat further with the material, the book also contains seven 'projects', which could serve as practical sessions related to one or more chapters.

The contributions focus on topics like Gröbner bases, real algebraic geometry, Lie algebras, factorisation of polynomials, integer programming, permutation groups, differential equations, coding theory, automatic theorem proving, and polyhedral geometry.

This book is a must-read for everybody interested in computer algebra.

目录

Chapter 1. GrSbner Bases, an Introduction

Arjeh M. Cohen

1. Introduction

2. Monomials

3. The Buchberger Algorithm

4. Standard Monomials

5. Solving Polynomial Equations

6. Effectiveness of Polynomial Rings

Chapter 2. Symbolic Recipes for Polynomial System Solving

Laureano Gonzalez-Vega, Fabrice Rouillier,and Marie-Frangoise Roy

1. Introduction

2. General Systems of Equations

2.1 Algebraic Preliminaries

2.2 First Recipes for Polynomial System Solving

3. Linear Algebra, Traces, and Polynomial Systems

3.1 Eigenvalues and Polynomial Systems

3.2 Counting Solutions and Removing Multiplicities

3.3 Rational Univariate Representation

4. As Many Equations as Variables

4.1 Generalities on Complete Intersection Polynomial Systems

4.2 Recipes for Polynomial System Solving When the Number of Equations Equals the Number of Unknowns

5. GrSbner Bases and Numerical Approximations

Chapter 3. Lattice Reduction

Frits Beukers

1. Introduction

2. Lattices

3. Lattice Reduction in Dimension 2

4. Lattice Reduction in Any Dimension

5. Implementations of the LLL-Algorithm

6. Small Linear Forms

Chapter 4. Factorisation of Polynomials

Frits Beukers

1. Introduction

2. Berlekamp's Algorithm

3. Additional Algorithms

4. Polynomials with Integer Coefficients

5. Factorisation of Polynomials with Integer Coefficients, I

6. Factorisation of Polynomials with Integer Coefficients, II

7. Factorisation in K[X], K Algebraic Number Field

Chapter 5. Computations in Associative and Lie Algebras Ggbor Ivanyos and Lajos Rdnyai

1. Introduction

2. Basic Definitions and Structure Theorems

3. Computing the Radical

4. Applications to Lie Algebras

5. Finding the Simple Components of Semisimple Algebras

6. Zero Divisors in Finite Algebras

Chapter 6. Symbolic Recipes for Real Solutions

Laureano Gonzalez-Vega, Fabrice Rouillier, Marie-Frangoise Roy,and Guadalupe Trujillo

1. Introduction

2. Real Root Counting: The Univariate Case.

2.1 Computing the Number of Real Roots

2.2 Sylvester Sequence

2.3 Sylvester-Habicht Sequence

2.4 Some Recipes for Counting Real Roots

3. Real Root Counting: The Multivariate Case

4. Tile Sign Determination Scheme

5. Real Algebraic Numbers and Thorn Codes

6. Quantifier Elimination

7. Appendix: Properties of the Polynomials in the Sylvester-Habicht Sequence

7.1 Definition and the Structure Theorem

7.2 Proof of the Structure Theorem

7.3 Sylvester-Habicht Sequences and Cauchy Index

Chapter 7. GrSbner Bases and Integer Programming Giinter M. Ziegler

1. Introduction

2. What is Integer Programming?

3. A Buchberger Algorithm for Integer Programming

……

Chapter 8. Working with Finite Groups

Chapter 9. Symbolic Analysis of Differential Equations

Chapter 10. Grobner Bases for Codes

Chapter 11. Grobner Bases for Decoding

Project 1. Automatic Geometry Theorem Proving

Project 2. The Birkhoff INterpolation Problem

Project 3. The Inverse Kinematics Problem in Robotics

Project 4. Quaternion Algebras

Project 5. Explorations with the Icosahedral Group

Project 6. The Small Mathieu Groups

Project 7. The Golay Codes

Index

 
 
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