Cohen Computer Algebra and Symbolic Computation and Symbolic Algebra Computer JOEL S. COHEN JOEL S. COHEN Computer Algebra and Symbolic Computation Computer Algebra and Mathematical Methods Symbolic Computation Mathematica™, Maple™, and similar software packages provide Mathematical Methods programs that carry out sophisticated mathematical operations. In this book the author explores the mathematical methods that form the basis for such programs, in particular the application of algorithms to methods such as automatic simplification, polynomial decomposition, and polynomial factorization. Computer Algebra and Symbolic Computation: Mathematical Methods goes beyond the basics of computer algebra—presented in Computer Algebra and Symbolic Computation: Elementary Algorithms—to explore complexity analysis of algorithms and recent developments in the field. This text: • is well-suited for self-study and can be used as the basis for a graduate course. • maintains the style set by Elementary Algorithms and explains mathematical methods as needed. • introduces advanced methods to treat complex operations. • presents implementations in such programs as Mathematica™ and Maple™. Mathematical Methods • includes a CD with the complete text, hyperlinks, and algorithms as well as additional reference files. For the student, Mathematical Methods is an essential companion to Elementary Algorithms, illustrating applications of basic ideas. For the professional, Mathematical Methods is a look at new applications of familiar concepts. ISBN 1-56881-159-4 A K A K Peters, Ltd. ì<(sl&q)=ibbfji< +^-Ä-U-Ä-U PETERS Computer Algebra and Symbolic Computation Computer Algebra and Symbolic Computation Mathematical Methods Joel S. Cohen Department of Computer Science University of Denver A K Peters Natick, Massachusetts Editorial, Sales, and Customer Service Office A K Peters, Ltd. 63 South Avenue Natick, MA 01760 www.akpeters.com Copyright © 2003 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Cohen, Joel S. Computer algebra and symbolic computation : mathematical methods / Joel S. Cohen p. cm. Includes bibliographical references and index. ISBN 1-56881-159-4 1. AlgebraData processing. I. Title. QA155.7.E4 .C635 2002 512dc21 2002024315 Printed in Canada 07 06 05 04 03 10 9 8 7 6 5 4 3 2 1 For my wife Kathryn vii Contents 1Preface ix 1 Background Concepts 1 1.1ComputerAlgebraSystems.................... 1 1.2 Mathematical Pseudo-Language (MPL) . 2 1.3 Automatic Simplification and Expression Structure . 5 1.4 General Polynomial Expressions . 11 1.5MiscellaneousOperators ..................... 12 2 Integers, Rational Numbers, and Fields 17 2.1TheIntegers............................ 17 2.2RationalNumberArithmetic................... 37 2.3Fields................................ 44 3 Automatic Simplification 63 3.1 The Goal of Automatic Simplification . 63 3.2 An Automatic Simplification Algorithm . 91 4 Single Variable Polynomials 111 4.1 Elementary Concepts and Polynomial Division . 111 4.2 Greatest Common Divisors in F[x] . 126 4.3 Computations in Elementary Algebraic Number Fields . 146 4.4PartialFractionExpansioninF(x)................ 166 viii 5 Polynomial Decomposition 179 5.1TheoreticalBackground.................... 180 5.2ADecompositionAlgorithm ................. 188 6 Multivariate Polynomials 201 6.1 Multivariate Polynomials and Integral Domains . 201 6.2 Polynomial Division and Expansion . 207 6.3GreatestCommonDivisors.................. 229 7TheResultant 265 7.1TheResultantConcept.................... 265 7.2 Polynomial Relations for Explicit Algebraic Numbers . 289 8 Polynomial Simplification with Side Relations 297 8.1MultipleDivisionandReduction............... 297 8.2 Equivalence, Simplification, and Ideals . 318 8.3ASimplificationAlgorithm..................334 9 Polynomial Factorization 349 9.1 Square-Free Polynomials and Factorization . 350 9.2 Irreducible Factorization: The Classical Approach . 360 9.3 Factorization in Zp[x]..................... 370 9.4 Irreducible Factorization: A Modern Approach . 399 Bibliography 431 Index 441 Q rshpr ]B8}Z sjAs $R Y hjI BP 8sY8s${R stI {B8}Z R{$t{ Ys $R {Bt{tI b$Y Y I%jB}8t $8}j8ts$Bt stI s}}j${s$Bt BP sjBT $Y8R Ys 8st$}Zjs stI stsjk9 8sY8s${sj -}RR$BtR qY$R ABB, stI Y {B8}st$Bt - 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