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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| Format: | eBook |
| Language: | English |
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Vienna
Springer Vienna
1996, 1996
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| Edition: | 1st ed. 1996 |
| Series: | Texts & Monographs in Symbolic Computation, A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria
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| 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 quantifierelimination
- 9.2 Cylindrical algebraic decomposition
- 9.3 Bibliographic notes
- 10 Indefinite summation