Polynomial Algorithms in Computer Algebra

283 Pages·1996·4.58 MB·English

by  Dipl.-Ing. Dr. Franz Winkler (auth.)

Texts and Monographs in
Symbolic Computation
A Series of the
Research Institute for Symbolic Computation,
Johannes-Kepler-University, linz, Austria
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, specif
ically 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 mo
tivation for studying computer algebra is given, and several prerequisites for
the area, such as algebraic preliminaries, representation of algebraic structures,
and complexity measurement are introduced. Some of the more important basic
domains of computer algebra are investigated in Chap. 2. Of course, this list is
by no means exhaustive. So, for instance, power series and matrices have not
been included in the list. The criterion for including a particular basic domain
was its importance for the more advanced topics in the subsequent chapters.
Computation by homomorphic images is presented in Chap. 3. Such homomor
phic images will be of great importance in gcd computation and factorization
of polynomials. These topics are dealt with in Chaps. 4 and 5. Chapter 6 con
tains algorithms for decomposition of polynomials. Linear systems appear often
as subproblems in different areas of computer algebra. They are investigated in
Chap. 7. Problems like computation of resultants, gcds, or factorizations of poly
nomials can be reduced to certain linear systems, so-called Hankel systems. In
Chap. 8 an introduction to the theory of Grabner bases for polynomial ideals is
given, and Grabner bases are applied to some important problems in polynomial
VI
ideal theory and solution of systems of polynomial equations. In the last three
chapters, polynomial algorithms are applied to some higher level problems in
computer algebra. Problems in real algebraic geometry can be decided by decid
ing problems in the elementary theory of real closed fields, i.e., by polynomial
algorithms. Such a decision algorithm is presented in Chap. 9. Chapter 10 gives
a description of Gosper's algorithm for solving summation problems. Finally, in
Chap. 11, gcd computation, factorization, and solution of systems of algebraic
equations are applied for deriving an algorithm for deciding whether an alge
braic curve can be parametrized by rational functions, and if so for computing
such a parametrization.
Clearly there are important topics in computer algebra missing from the
contents of this book, such as simplification of expressions, integration of ele
mentary functions, computer algebra solutions to differential equation problems,
or algebraic computations in finite group theory. Including all these other top
ics would increase the size of the book beyond any reasonable bound. For this
reason I limit myself to discussing that part of computer algebra, which deals
with polynomials.
In recent years several books on computer algebra have been published. They
all approach the field from their own particular angle. The emphasis in this book
is on introducing polynomial algorithms in computer algebra from the bottom
up, starting from very basic problems in computation over the integers, and
finally leading to, e.g., advanced topics in factorization, solution of polynomial
equations and constructive algebraic geometry. Along the way, the complexity
of many of the algorithms is investigated.
I hope that this book might serve as the basis for exciting new developments
in computer algebra.
Franz Winkler
Contents
1 Introduction 1
1.1 What is computer algebra?
1.2 Program systems in computer algebra 8
1.3 Algebraic preliminaries 13
1.4 Representation of algebraic structures 18
1.5 Measuring the complexity of algorithms 22
1.6 Bibliographic notes 24
2 Arithmetic in basic domains 26
2.1 Integers 26
2.2 Polynomials 37
2.3 Quotient fields 43
2.4 Algebraic extension fields 45
2.5 Finite fields 47
2.6 Bibliographic notes 50
3 Computing by homomorphic images 51
3.1 The Chinese remainder problem and the modular method 51
3.2 p-adic approximation 61
3.3 The fast Fourier transform 67
3.4 Bibliographic notes 81
4 Greatest common divisors of polynomials 82
4.1 Polynomial remainder sequences 82
4.2 A modular gcd algorithm 91
4.3 Computation of resultants 96
4.4 Squarefree factorization 101
4.5 Squarefree partial fraction decomposition 104
4.6 Integration of rational functions 105
4.7 Bibliographic notes 111
5 Factorization of polynomials 112
5.1 Factorization over finite fields 112
5.2 Factorization over the integers 115
5.3 A polynomial-time factorization algorithm over the integers 125
5.4 Factorization over algebraic extension fields 140
viii
5.5 Factorization over an algebraically closed field 146
5.6 Bibliographic notes ISO
6 Decomposition of polynomials 151
6.1 A polynomial-time algorithm for decomposition lSI
6.2 Bibliographic notes 156
7 Linear algebra - solving linear systems 157
7.1 Bareiss's algorithm 157
7.2 Hankel matrices 163
7.3 Application of Hankel matrices to polynomial problems 166
7.4 Bibliographic notes 171
8 The method of Grobner bases 172
8.1 Reduction relations 172
8.2 Polynomial reduction and Grabner bases 179
8.3 Computation of Grabner bases 183
8.4 Applications of Grabner bases 189
8.5 Speed-ups and complexity considerations 201
8.6 Bibliographic notes 202
9 Quantifier elimination in real closed fields 204
9.1 The problem of quantifier elimination 204
9.2 Cylindrical algebraic decomposition 206
9.3 Bibliographic notes 214
10 Indefinite summation 215
10.1 Gosper's algorithm 215
10.2 Bibliographic notes 223
11 Parametrization of algebraic curves 224
11.1 Plane algebraic curves 224
11.2 A parametrization algorithm 230
11.3 Bibliographic notes 236
Solutions of selected exercises 237
References 249
Subject index 265
1
Introduction
1.1 What is computer algebra?
In the recent decades it has been more and more realized that computers are
of enormous importance for numerical computations. However, these powerful
general purpose machines can also be used for transforming, combining and
computing symbolic algebraic expressions. In other words, computers can not
only deal with numbers, but also with abstract symbols representing mathemat
ical formulas. This fact has been realized much later and is only now gaining
acceptance among mathematicians and engineers.
Mathematicians in the old period, say before 1850 A.D., solved the majority
of mathematical problems by extensive calculations. A typical example of this
type of mathematical problem solver is Euler. Even Gauss in 180 I temporarily
abandoned his research in arithmetic and number theory in order to calculate
the orbit of the newly discovered planetoid Ceres. It was this calculation much
more than his masterpiece Disquisitiones Arithmeticae which became the basis
for his fame as the most important mathematician of his time.
So it is not astonishing that in the 18th and beginning 19th centuries many
mathematicians were real wizzards of computation. However, during the 19th
century the style of mathematical research changed from quantitative to qual
itative aspects. A number of reasons were responsible for this change, among
them the importance of providing a sound basis for the vast theory of analysis.
But the fact that computations gradually became more and more complicated
certainly also played its role. This impediment has been removed by the advent
of modern digital computers in general and by the development of program
systems in computer algebra, in particular. By the aid of computer algebra the
capacity for mathematical problem solving has been decisively improved.
Even in our days many mathematicians think that there is a natural division
of labor between man and computer: a person applies the appropriate algebraic
transformations to the problem at hand and finally arrives at a program which
then can be left to a "number crunching" computer. But already in 1844, Lady
Augusta Ada Byron, countess Lovelace, recognized that this division of labor
is not inherent in mathematical problem solving and may be even detrimental.
In describing the possible applications of the Anal.vtical Engine developed by
Charles Babbage she writes:
"Many persons who are not conversant with mathematical studies imagine that
because the business of [Babbage's Analytical Engine] is to give its results in
numerical notation, the nature of its processes must consequently be arithmetical
2 Introduction
and numerical rather than algebraic and analytical. This is an error. The engine
can arrange and combine its numerical quantities exactly as if they were letters
or any other general symbols; and in fact it might bring out its results in algebraic
notation were provisions made accordingly."
And indeed a modem digital computer is a "universal" machine capable of
carrying out an arbitrary algorithm, i.e., an exactly specified procedure, algebraic
algorithms being no exceptions.
An attempt at a definiton
Now what exactly is symbolic algebraic computation or, in other words, com
puter algebra? In his introduction to Buchberger et al. (1983), R. Loos made
the following attempt at a definition:
"Computer algebra is that part of computer science which designs, analyzes,
implements, and applies algebraic algorithms."
While it is arguable whether computer algebra is part of computer science or
mathematics, we certainly agree with the rest of the statement. In fact, in our
view computer algebra is a special form of scientific computation, and it com
prises a wide range of basic goals, methods, and applications. In contrast to
numerical computation the emphasis is on computing with symbols represent
ing mathematical concepts. Of course that does not mean that computer algebra
is devoid of computations with numbers. Decimal or other positional represen
tations of integers, rational numbers and the like appear in any symbolic com
putation. But integers or real numbers are not the sole objects. In addition to
these basic numerical entities, computer algebra deals with polynomials, rational
functions, trigonometric functions, algebraic numbers, etc. That does not mean
that we will not need numerical algorithms any more. Both forms of scientific
computation have their merits and they should be combined in a computational
environment. For instance, in order to compute an approximate solution to a
differential equation it might be reasonable to determine the first n terms of a
power series solution by exact methods from computer algebra before handing
these terms over to a numerical package for evaluating the power series.
Summarizing, we might list the following characteristics of computer alge
bra:
1. Computer algebra is concerned with computing in algebraic structures.
This might mean in basic algebraic number domains, in algebraic extensions of
such domains, in polynomial rings or function fields, in differential or differ
ence fields, in the abstract setting of group theory, or the like. Often it is more
economical in terms of computation time to simplify an expression algebraically
before evaluating it numerically. In this way the expression becomes simpler
and less prone to numerical errors.
2. The results of computer algebra algorithms are exact and not subject
to approximation errors. So, typically, when we solve a system of algebraic