AN A K PETERS BOOK Computer Algebra and Symbolic Computation Q ~ Computer Algebra and Symbolic Computation Mathematical Methods Joel S. Cohen Department of Computer Science University of Denver Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business AN A K PETERS BOOK First published 2003 by A K Peters, Ltd. Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2003 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13: 978-1-56881-159-8 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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QAI55.7.E4 .C6352 2002 512-dc21 2002024315 For my wife Kathryn Q ~ Contents Preface ix 1 Background Concepts 1 1.1 Computer Algebra Systems 1 1.2 Mathematical Pseudo-Language (MPL) 2 1.3 Automatic Simplification and Expression Structure .... 5 1.4 General Polynomial Expressions 11 1.5 Miscellaneous Operators 12 2 Integers, Rational Numbers, and Fields 17 2.1 The Integers 17 2.2 Rational Number Arithmetic 37 2.3 Fields 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 Ill 4.2 Greatest Common Divisors in F[x] 126 4.3 Computations in Elementary Algebraic Number Fields . 146 4.4 Partial Fraction Expansion in F(x) 166 vii viii Contents 5 Polynomial Decomposition 179 5.1 Theoretical Background 180 5.2 A Decomposition Algorithm 188 6 Multivariate Polynomials 201 6.1 Multivariate Polynomials and Integral Domains 201 6.2 Polynomial Division and Expansion 207 6.3 Greatest Common Divisors 229 7 The Resultant 265 7.1 The Resultant Concept 265 7.2 Polynomial Relations for Explicit Algebraic Numbers .... 289 8 Polynomial Simplification with Side Relations 297 8.1 Multiple Division and Reduction 297 8.2 Equivalence, Simplification, and Ideals 318 8.3 A Simplification Algorithm 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 Preface Computer algebra is the field of mathematics and computer science that is concerned with the development, implementation, and application of algo­ rithms that manipulate and analyze mathematical expressions. This book and the companion text, Computer Algebra and Symbolic Computation: Elementary Algorithms, are an introduction to the subject that addresses both its practical and theoretical aspects. Elementary Algorithms addresses the practical side; it is concerned with the formulation of algorithms that solve symbolic mathematical problems and with the implementation of these algorithms in terms of the operations and control structures available in computer algebra programming languages. This book, which addresses more theoretical issues, is concerned with the basic mathematical and algo­ rithmic concepts that are the foundation of the subject. Both books serve as a bridge between texts and manuals that show how to use computer algebra software and graduate level texts that describe algorithms at the forefront of the field. These books have been in various stages of development for over 15 years. They are based on the class notes for a two-quarter course sequence in computer algebra that has been offered at the University of Denver every other year for the past 16 years. The first course, which is the basis for El­ ementary Algorithms, attracts primarily undergraduate students and a few graduate students from mathematics, computer science, and engineering. The second course, which is the basis for Mathematical Methods, attracts primarily graduate students in both mathematics and computer science. The course is cross-listed under both mathematics and computer science. IX X Preface Prerequisites The target audience for these books includes students and professionals from mathematics, computer science, and other technical fields who would like to know about computer algebra and its applications. In the spirit of an introductory text, we have tried to minimize the prerequisites. The mathematical prerequisites include the usual two year freshman-sophomore sequence of courses (calculus through multivariable calculus, elementary linear algebra, and applied ordinary differential equa­ tions). In addition, an introductory course in discrete mathematics is rec­ ommended because mathematical induction is used as a proof technique throughout. Topics from elementary number theory and abstract algebra are introduced as needed. On the computer science side, we assume that the reader has had some experience with a computer programming language such as Fortran, Pascal, C, C++, or Java. Although these languages are not used in these books, the skills in problem solving and algorithm development obtained in a be­ ginning programming course are essential. One programming technique that is especially important in computer algebra is recursion. Although many students will have seen recursion in a conventional programming course, the topic is described in Chapter 5 of Elementary Algorithms from a computer algebra perspective. Realistically speaking, while these prerequisites suffice in a formal sense for both books, in a practical sense there are some sections as the texts progress where greater mathematical and computational sophistication is required. Although the mathematical development in these sections can be challenging for students with the minimum prerequisites, the algorithms are accessible, and these sections provide a transition to more advanced treatments of the subject. Organization and Content Broadly speaking, these books are intended to serve two (complementary) purposes: • To provide a systematic approach to the algorithmic formulation and implementation of mathematical operations in a computer algebra programming language. Algorithmic methods in traditional mathematics are usually not pre­ sented with the precision found in numerical mathematics or conventional computer programming. For example, the algorithm for the expansion of products and powers of polynomials is usually given informally instead of with (recursive) procedures that can be expressed as a computer program. Preface XI The material in Elementary Algorithms is concerned with the algorith­ mic formulation of solutions to elementary symbolic mathematical prob­ lems. The viewpoint is that mathematical expressions, represented as ex­ pression trees, are the data objects of computer algebra programs, and, using a few primitive operations that analyze and construct expressions, we can implement many elementary operations from algebra, trigonometry, calculus, and differential equations. For example, algorithms are given for the analysis and manipulation of polynomials and rational expressions, the manipulation of exponential and trigonometric functions, differentiation, elementary integration, and the solution of first order differential equa­ tions. Most of the material in this book is not found in either mathematics textbooks or in other, more advanced computer algebra textbooks. • To describe some of the mathematical concepts and algorithmic tech- niques utilized by modern computer algebra software. For the past 35 years, the research in computer algebra has been con­ cerned with the development of effective and efficient algorithms for many mathematical operations including polynomial