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
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245 0 0 |a Polynomial Algorithms in Computer Algebra  |h Elektronische Ressource  |c by Franz Winkler 
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260 |a Vienna  |b Springer Vienna  |c 1996, 1996 
300 |a VIII, 270 p  |b online resource 
505 0 |a 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 
505 0 |a 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 --  
505 0 |a 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 --  
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653 |a Algorithms 
653 |a Theory of Computation 
653 |a Algorithms 
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653 |a Symbolic and Algebraic Manipulation 
653 |a Algebra 
653 |a Algebra 
653 |a Data structures (Computer science) 
653 |a Data Structures 
653 |a Algorithm Analysis and Problem Complexity 
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520 |a 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 set of course notes has emerged from these activities. People have asked me for copies of the course notes, and different versions of them have been circulating for a few years. Finally I decided that I should really take the time to write the material up in a coherent way and make a book out of it. Here, now, is the result of this work. Over the years many students have been helpful in improving the quality of the notes, and also several colleagues at Linz and elsewhere have contributed to it. I want to thank them all for their effort, in particular I want to thank B. Buchberger, who taught me the theory of Grabner bases nearly two decades ago, B. F. Caviness and B. D. Saunders, who first stimulated my interest in various problems in computer algebra, G. E. Collins, who showed me how to compute in algebraic domains, and J. R. Sendra, with whom I started to apply computer algebra methods to problems in algebraic geometry. Several colleagues have suggested improvements in earlier versions of this book. However, I want to make it clear that I am responsible for all remaining mistakes