Some tapas of computer algebra.計算機代數(shù)一些Tapas

內(nèi)容概要

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.

書籍目錄

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 RingsChapter 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 ApproximationsChapter 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 FormsChapter 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 FieldChapter 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 AlgebrasChapter 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 IndexChapter 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 GroupsChapter 9. Symbolic Analysis of Differential EquationsChapter 10. Grobner Bases for CodesChapter 11. Grobner Bases for DecodingProject 1. Automatic Geometry Theorem ProvingProject 2. The Birkhoff INterpolation ProblemProject 3. The Inverse Kinematics Problem in RoboticsProject 4. Quaternion AlgebrasProject 5. Explorations with the Icosahedral GroupProject 6. The Small Mathieu GroupsProject 7. The Golay CodesIndex

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