Polynomial Algorithms in Computer Algebra

For several years now I have been teaching courses in computer algebra at the Universitat Linz, the University of Delaware, and the Universidad de Alcala de Henares. In the summers of 1990 and 1992 I have organized and taught summer schools in computer algebra at the Universitat Linz. Gradually a se...

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Main Author: Winkler, Franz
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Vienna Springer Vienna 1996, 1996
Edition:1st ed. 1996
Series:Texts & Monographs in Symbolic Computation, A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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