Cohen

Computer Algebra and
Symbolic Computation
Mathematical Methods

Mathematica™, Maple™, and similar software packages provide
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:

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 Peters, Ltd.

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Mathematical Methods

• 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™.
• includes a CD with the complete text, hyperlinks, and algorithms
as well as additional reference files.

Computer Algebra and Symbolic Computation

JOEL S. COHEN

AK
PETERS

JOEL S. COHEN

Computer Algebra and
Symbolic Computation
Mathematical Methods


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Computer Algebra and Symbolic Computation

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Computer Algebra and Symbolic Computation
Mathematical Methods

Joel S. Cohen
Department of Computer Science
University of Denver

A K Peters
Natick, Massachusetts

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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. Algebra–Data processing. I. Title.
QA155.7.E4 .C635 2002
512–dc21

Printed in Canada
07 06 05 04 03

2002024315

10 9 8 7 6 5 4 3 2 1

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For my wife Kathryn

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vii

Contents

1 Preface

ix

1 Background Concepts
1.1 Computer Algebra Systems . . . . . . . . . . . . .
1.2 Mathematical Pseudo-Language (MPL) . . . . . .
1.3 Automatic Simplification and Expression Structure
1.4 General Polynomial Expressions . . . . . . . . . .
1.5 Miscellaneous Operators . . . . . . . . . . . . . .

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1
1
2
5
11
12

2 Integers, Rational Numbers, and Fields
2.1 The Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Rational Number Arithmetic . . . . . . . . . . . . . . . . . . .
2.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
17
37
44

3 Automatic Simplification
3.1 The Goal of Automatic Simplification . . . . . . . . . . . . . .
3.2 An Automatic Simplification Algorithm . . . . . . . . . . . . .

63
63
91

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4 Single Variable Polynomials
4.1 Elementary Concepts and Polynomial Division . . . . . . . .
4.2 Greatest Common Divisors in F[x] . . . . . . . . . . . . . . .
4.3 Computations in Elementary Algebraic Number Fields . . .
4.4 Partial Fraction Expansion in F(x) . . . . . . . . . . . . . . .

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111
111
126
146
166


viii

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
9.1 Square-Free Polynomials and Factorization . . . .
9.2 Irreducible Factorization: The Classical Approach
9.3 Factorization in Zp [x] . . . . . . . . . . . . . . . .
9.4 Irreducible Factorization: A Modern Approach . .

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349
350
360
370
399

Bibliography

431

Index

441

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1
Background Concepts

In this chapter we summarize the background material that provides a
framework for the mathematical and computational discussions in the book.
A more detailed discussion of this material can be found on the CD that
accompanies this book and in our companion book, Computer Algebra and
Symbolic Computation, Elementary Algorithms, (Cohen [24]). Readers who
are familiar with this material may wish to skim this chapter and refer to
it as needed.

1.1 Computer Algebra Systems
A computer algebra system (CAS) or symbol manipulation system is a computer program that performs symbolic mathematical operations. In this
book we refer to the computer algebra capabilities of the following three
systems which are readily available and support a programming style that
is most similar to the one used here:

• Maple – a very large CAS originally developed by the Symbolic
Computation Group at the University of Waterloo (Canada) and now
distributed by Waterloo Maple Inc. Information about Maple is found
in Heck [45] or at the web site .
• Mathematica – a very large CAS developed by Wolfram Research
Inc. Information about Mathematica can be found in Wolfram [102]
or at the web site .
1

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2

1. Background Concepts

• MuPAD – a large CAS developed by the University of Paderborn
(Germany) and SciFace Software GmbH & Co. KG. Information about
MuPAD can be found in Gerhard et al. [40] or at the web site
.

1.2 Mathematical Pseudo-Language (MPL)
Mathematical pseudo-language (MPL) is an algorithmic language that is
used throughout this book to describe the concepts, examples, and algorithms of computer algebra. MPL algorithms are readily expressed in the
programming languages of Maple, Mathematica, and MuPAD, and implementations of the dialogues and algorithms in these systems are included
on the CD that accompanies this book.

Mathematical Expressions
MPL mathematical expressions are constructed with the following symbols
and operators:

• Integers and fractions that utilize infinite precision rational number
arithmetic.
• Identifiers that are used both as programming variables that represent the result of a computation and as mathematical symbols that
represent indeterminates (or variables) in a mathematical expression.
• The algebraic operators +, −, ∗, /, ∧ (power), and ! (factorial). (As
with ordinary mathematical notation, we usually omit the ∗ operator
and use raised exponents for powers.)
• Function forms that are used for mathematical functions (sin(x),
exp(x), arctan(x), etc.), mathematical operators (Expand(u), Factor(u), Integral(u,x), etc.), and undefined functions (f(x), g(x,y), etc.).
• The relational operators =, =, <, ≤, >, and ≥, the logical constants
true and false, and the logical operators, and, or, and not.
• Finite sets of expressions that utilize the set operations ∪, ∩, ∼ (set
difference), and ∈ (set membership). Following mathematical convention, sets do not contain duplicate elements and the contents of a set
does not depend on the order of the elements (e.g., {a, b} = {b, a}).
• Finite lists of expressions. A list is represented using the brackets [
and ] (e.g., [1, x, x2 ]). The empty list, which contains no expressions,

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1.2. Mathematical Pseudo-Language (MPL)

3

is represented by [ ]. Lists may contain duplicate elements, and the
order of elements is significant (e.g., [a, b] = [b, a]).
The MPL set and list operators and the corresponding operators in
computer algebra systems are given in Figure 1.1.
MPL mathematical expressions have two (somewhat overlapping) roles
as either program statements that represent a computational step in a program or as data objects that are processed by program statements.

Assignments, Functions, and Procedures
The MPL assignment operator is a colon followed by an equal sign (:=) and
an assignment statement has the form f := u where u is a mathematical
expression.
function

An MPL function definition has the form f (x1 , . . . , xl ) := u, where
x1 , . . . , xl is a sequence of symbols called the formal parameters, and u is
a mathematical expression. MPL procedures extend the function concept
to mathematical operators that are defined by a sequence of statements.
The general form of an MPL procedure is given in Figure 1.2. Functions
and procedures are invoked with an expression of the form f (a1 , . . . , al ),
where a1 , . . . , al is a sequence of mathematical expressions called the actual
parameters.
In order to promote a programming style that works for all languages,
we adopt the following conventions for the use of local variables and formal
parameters in a procedure:
• An unassigned local variable cannot appear as a symbol in a mathematical expression. In situations where a procedure requires a local (unassigned) mathematical symbol, we either pass the symbol
through the parameter list or use a global symbol.
• Formal parameters are used only to transmit data into a procedure
and not as local variables or to return data from a procedure. When
we need to return more than one expression from a procedure, we
return a set or list of expressions.

Decision and Iteration Structures
MPL provides three decision structures: the if structure, the if-else structure which allows for two alternatives, and the multi-branch decision structure which allows for a sequence of alternatives.
MPL contains two iteration structures that allow for repeated evaluation of a sequence of statements, the while structure and the for structure.
Some of our procedures contain for loops that include a Return statement.

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4

1. Background Concepts
MPL
set notation
{a, b, c}
∅
A∪B
(set union)
A∩B
(set intersection)
A∼B
(set difference)
x∈A
(set membership)

Maple

Mathematica

MuPAD

{a,b,c}
{ }
A union B

{a,b,c}

{ }
Union[A,B]

{a,b,c}
{ }
A union B

A intersect B

Intersection[A,B]

A intersect B

A minus B

Complement[A,B]

A minus B

member(x, A)

MemberQ[x,A]

contains(A,x)

(a) Sets. (Implementation: Maple (mws), Mathematica (nb), MuPAD (mnb).)

MPL
list notation
[a, b, c]

empty list [ ]
First (L)
(first member
of L)
Rest(L)
(a new list
with first
member of
L removed)
Adjoin(x, L)
(a new list
with x
adjoined to
the beginning
of L)
Join (L, M )
(a new list
with members
of L followed
by members
of M )
x∈L
(list
membership)

Maple

Mathematica

MuPAD

[a,b,c]
[]
op(1,L)

{a,b,c}
{}
First[L]

[a,b,c]
[]
op(L,1)

[op(2..nops(L),L)]

Rest[L]

[op(L,2..nops(L))]

[x,op(L)]

Prepend[L,x]

[x, op(L)]

[op(L),op(M)]

Join[L,M]

concat(L,M)

member(x,L)

MemberQ[x,L]

contains(L,x)

(b) Lists. (Implementation: Maple (mws), Mathematica (nb), MuPAD (mnb).)

Figure 1.1. MPL set and list operations in CAS languages.

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1.3. Automatic Simplification and Expression Structure

5

Procedure f (x1 , . . . , xl );
Input
x1 : description of input to x1 ;
..
.
xl : description of input to xl ;
Output
description of output;
Local Variables
v1 , . . . , vm ;
Begin
S1 ;

..
.
Sn
End

Figure 1.2. The general form of an MPL procedure. (Implementation: Maple
(txt), Mathematica (txt), MuPAD (txt).)
In this case, we intend that both the loop and the procedure terminate
when the Return is encountered.1
All computer algebra languages provide decision and iteration structures (Figure 1.3).

1.3 Automatic Simplification and Expression Structure
As part of the evaluation process, computer algebra systems apply some
“obvious” simplification rules from algebra and trigonometry that remove
extraneous symbols from an expression and transform it to a standard form.
This process is called automatic simplification. For example,
√
x + 2 x + y y 2 + z 0 + sin(π/4) → 3 x + y 3 + 1 + 2/2
where the expression to the right of the arrow gives the automatically
simplified form after evaluation.
In MPL (as in a CAS), all expressions in dialogues and computer programs operate in the context of automatic simplification. This means: (1)
1 The for statements in both Maple and MuPAD work in this way. However, in
Mathematica, a Return in a For statement will only work in this way if the upper limit
contains a relational operator (e.g., i<=N). (Implementation: Mathematica (nb).)

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