Greatest common divisors of polynomials
If K is a field, then K[x] is a Euclidean domain, so gcd(f, g) for f, g ∈ K[x] can be computed by the Euclidean algorithm. Often, however, we are given polynomials f, g over a domain such as ℤ or K[x1, …, xn-1] and we need to compute their gcd.
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Edited by B. Buchberger and G. E. Collins
F. Winkler Polynomial Algorithms in Computer Algebra
Springer-Verlag Wien GmbH
Dipl.-Ing. Df. Franz Winkler Research Institute for Symbolic Computation Johannes-Kepler-University Linz, Linz, Austria
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photo-copying machi nes or similar means, and storage in data banks. © 1996 Springer-Verlag Wien Originally published by Springer-VerlaglWien in 1996 Data con vers ion by H.-D. Ecker, Btiro ftir Textverarbeitung, Bonn Printed on acid-free and chlorine-free bleached paper
With 13 Figures Library of Congress Cataloging-in-Publication Data Winkler, Franz. Polynomial algorithms in computer algebra / Franz Winkler. p. cm. - (Texts and monographs in symbolic computation, ISSN 0943-853X) Includes bibliographical references and index. ISBN 978-3-211-82759-8 ISBN 978-3-7091-6571-3 (eBook) DOI 1O.l007/978-3-7091-6571-3 1. Algebra-Data processing. 2. Computer algorithms. I. Title. II. Series. QAI55.7.E4W56 1996 512.9'42-dc20 96-7170 CIP
ISSN 0943-853X
ISBN 978-3-211-82759-8
Preface 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. Research of the author was partially supported by Osterreichischer Fonds zur Farderung der wissenschaftlichen Forschung, project nos. P6763 (ASAG) and P8573 (SGC). Let me give a brief overview of the contents of this book. In Chap. I a motivation for studying computer algebra is given, and several prerequisites for the area, such as algebraic preliminaries, representation of algebraic structures, and complexity measu
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