Polynomial Algorithms in Computer Algebra

ISBN-13: 9783211827598 / Angielski / Miękka / 1996 / 270 str.

F. Winkler; Franz Winkler

Opis

The book gives a thorough introduction to the mathematical underpinnings of computer algebra. The subjects treated range from arithmetic of integers and polynomials to fast factorization methods, Grobner bases, and algorithms in algebraic geometry. The algebraic background for all the algorithms presented in the book is fully described, and most of the algorithms are investigated with respect to their computational complexity. Each chapter closes with a brief survey of the related literature."

Spis treści

1 Introduction.- 1.1 What is computer algebra?.- 1.2 Program systems in computer algebra.- 1.3 Algebraic preliminaries.- 1.4 Representation of algebraic structures.- 1.5 Measuring the complexity of algorithms.- 1.6 Bibliographic notes.- 2 Arithmetic in basic domains.- 2.1 Integers.- 2.2 Polynomials.- 2.3 Quotient fields.- 2.4 Algebraic extension fields.- 2.5 Finite fields.- 2.6 Bibliographic notes.- 3 Computing by homomorphic images.- 3.1 The Chinese remainder problem and the modular method.- 3.2 p-adic approximation.- 3.3 The fast Fourier transform.- 3.4 Bibliographic notes.- 4 Greatest common divisors of polynomials.- 4.1 Polynomial remainder sequences.- 4.2 A modular gcd algorithm.- 4.3 Computation of resultants.- 4.4 Squarefree factorization.- 4.5 Squarefree partial fraction decomposition.- 4.6 Integration of rational functions.- 4.7 Bibliographic notes.- 5 Factorization of polynomials.- 5.1 Factorization over finite fields.- 5.2 Factorization over the integers.- 5.3 A polynomial-time factorization algorithm over the integers.- 5.4 Factorization over algebraic extension fields.- 5.5 Factorization over an algebraically closed field.- 5.6 Bibliographic notes.- 6 Decomposition of polynomials.- 6.1 A polynomial-time algorithm for decomposition.- 6.2 Bibliographic notes.- 7 Linear algebra—solving linear systems.- 7.1 Bareiss’s algorithm.- 7.2 Hankel matrices.- 7.3 Application of Hankel matrices to polynomial problems.- 7.4 Bibliographic notes.- 8 The method of Gröbner bases.- 8.1 Reduction relations.- 8.2 Polynomial reduction and Gröbner bases.- 8.3 Computation of Gröbner bases.- 8.4 Applications of Gröbner bases.- 8.5 Speed-ups and complexity considerations.- 8.6 Bibliographic notes.- 9 Quantifier elimination in real closed fields.- 9.1 The problem of quantifier elimination.- 9.2 Cylindrical algebraic decomposition.- 9.3 Bibliographic notes.- 10 Indefinite summation.- 10.1 Gosper’s algorithm.- 10.2 Bibliographic notes.- 11 Parametrization of algebraic curves.- 11.1 Plane algebraic curves.- 11.2 A parametrization algorithm.- 11.3 Bibliographic notes.- Solutions of selected exercises.- References.