Ebook: 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 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.

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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, Gr?bner 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.

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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, Gr?bner 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.
Content:
Front Matter....Pages i-viii
Introduction....Pages 1-25
Arithmetic in basic domains....Pages 26-50
Computing by homomorphic images....Pages 51-81
Greatest common divisors of polynomials....Pages 82-111
5 Factorization of polynomials....Pages 112-150
Decomposition of polynomials....Pages 151-156
Linear algebra—solving linear systems....Pages 157-171
The method of Gr?bner bases....Pages 172-203
Quantifier elimination in real closed fields....Pages 204-214
10 Indefinite summation....Pages 215-223
Parametrization of algebraic curves....Pages 224-236
Back Matter....Pages 237-279

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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, Gr?bner 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.
Content:
Front Matter....Pages i-viii
Introduction....Pages 1-25
Arithmetic in basic domains....Pages 26-50
Computing by homomorphic images....Pages 51-81
Greatest common divisors of polynomials....Pages 82-111
5 Factorization of polynomials....Pages 112-150
Decomposition of polynomials....Pages 151-156
Linear algebra—solving linear systems....Pages 157-171
The method of Gr?bner bases....Pages 172-203
Quantifier elimination in real closed fields....Pages 204-214
10 Indefinite summation....Pages 215-223
Parametrization of algebraic curves....Pages 224-236
Back Matter....Pages 237-279
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