http://dx.doi.org/10.1090/gsm/003
Selected Titles in This Series
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8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996
7 Gerald J. Janusz, Algebraic number fields, second edition, 1996
6 Jens Carsten Jantzen, Lectures on quantum groups, 1996
5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995
4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994
3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994
2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993
1 Ethan Akin, The general topology of dynamical systems, 1993
An Introduction to
Grobner Bases
William W.Adams
Philippe Loustaunau
Graduate Studies
in Mathematics
Volum e 3
Ivfffl'J J America n Mathematica l Societ y
^^^
Editorial Board
James E. Humphreys
Robion C. Kirby
Lance Small
2000 Mathematics
Subject Classification.
Primary 13P10.
ABSTRACT. Grobner bases are the primary tool for doing explicit computations in polynomial rings
in many variables. In this book we give a leisurely introduction to the subject and its applications
suitable for students with a little knowledge of abstract and linear algebra. T h e book contains
not only the theory over fields, but also, the theory in modules and over rings.
Library of Congress Cataloging-in-Publication D a t a
Adams, William W., 1937An introduction to Grobner bases/William W. Adams, Philippe Loustaunau.
p. cm. — (Graduate studies in mathematics, ISSN 1065-7339; 3)
Includes bibliographical references and index.
ISBN 0-8218-3804-0
1. Grobner bases. I. Loustaunau, Philippe, 1958- . II. Title. III. Series.
QA251.3.A32
512 / .4—dc20
1994
94-19081
CIP
C o p y i n g and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of t h e material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Acquisitions Department, American Mathematical Society,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
e-mail to reprint-permissionQams.org.
© 1994 by the American Mathematical Society. All rights reserved.
Reprinted with corrections in 1996.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ T h e paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
This publication was typeset by the authors, with assistance
from the American Mathematical Society, using ^v^S-T^gX,
the American Mathematical Society's TfeX macro system.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4
15 14 13 12 11 10 09
To my wife Bizabeth
and our daughters Ruth and Sarah
WWA
To my wife Yvonne
and our children Bleen, Gareth, and Manon
PL
Contents
Preface
ix
Chapter 1. Basic Theory of Grobner Bases
1.1. Introduction
1.2. The Linear Case
1.3. The One Variable Case
1.4. Term Orders
1.5. Division Algorithm
1.6. Grobner Bases
1.7. S-Polynomials and Buchberger's Algorithm
1.8. Reduced Grobner Bases
1.9. Summary
1
1
7
10
18
25
32
39
46
50
Chapter 2. Applications of Grobner Bases
2.1. Elementary Applications of Grobner Bases
2.2. Hilbert Nullstellensatz
2.3. Elimination
2.4. Polynomial Maps
2.5. Some Applications to Algebraic Geometry
2.6. Minimal Polynomials of Elements in Field Extensions
2.7. The 3-Color Problem
2.8. Integer Programming
53
53
61
69
79
90
97
102
105
Chapter 3. Modules and Grobner Bases
3.1. Modules
3.2. Grobner Bases and Syzygies
3.3. Improvements on Buchberger's Algorithm
3.4. Computation of the Syzygy Module
3.5. Grobner Bases for Modules
3.6. Elementary Applications of Grobner Bases for Modules
3.7. Syzygies for Modules
3.8. Applications of Syzygies
3.9. Computation of Horn
113
113
118
124
134
140
152
161
171
183
Vll
CONTENTS
Vlll
3.10. Free Resolutions
194
Chapter 4. Grobner Bases over Rings
4.1. Basic Definitions
4.2. Computing Grobner Bases over Rings
4.3. Applications of Grobner Bases over Rings
4.4. A Primality Test
4.5. Grobner Bases over Principal Ideal Domains
4.6. Primary Decomposition in R[x] for R a PID
201
202
212
225
237
246
259
Appendix A. Computations and Algorithms
275
Appendix B. Well-ordering and Induction
277
References
279
List of Symbols
283
Index
Preface
We wrote this book with two goals in mind:
(i) To give a leisurely and fairly comprehensive introduction to the definition
and construction of Grobner bases;
(ii) To discuss applications of Grobner bases by presenting computational
methods to solve problems which involve rings of polynomials.
This book is designed to be a first course in the theory of Grobner bases suitable
for an advanced undergraduate or a beginning graduate student. This book is
also suitable for students of computer science, applied mathematics, and engineering who have some acquaintance with modern algebra. The book does not
assume an extensive knowledge of algebra. Indeed, one of the attributes of this
subject is that it is very accessible. In fact, all that is required is the notion of the
ring of polynomials in several variables (and rings in general in a few places, in
particular in Chapter 4) together with the ideals in this ring and the concepts of
a quotient ring and of a vector space introduced at the level of an undergraduate
abstract and linear algebra course. Except for linear algebra, even these ideas
are reviewed in the text. Some topics in the later sections of Chapters 2, 3, and 4
require more advanced material. This is always clearly stated at the beginning of
the section and references are given. Moreover, most of this material is reviewed
and basic theorems are stated without proofs.
The book can be read without ever "computing" anything. The theory stands
by itself and has important theoretical applications in its own right. However,
the reader will not fully appreciate the power of, or get insight into, the methods
introduced in the book without actually doing some of the computations in the
examples and the exercises by hand or, more often, using a Computer Algebra
System (there are over 120 worked-out examples and over 200 exercises). Computing is useful in producing and analyzing examples which illustrate a concept
already understood, or which one hopes will give insight into a less well understood idea or technique. But the real point here is that computing is the very
essence of the subject. This is why Grobner basis theory has become a major
research area in computational algebra and computer science. Indeed, Grobner
basis theory is generating increasing interest because of its usefulness in proIX
X
PREFACE
viding computational tools which are applicable to a wide range of problems in
mathematics, science, engineering, and computer science.
Grobner bases were introduced in 1965 by Bruno Buchberger 1 [Bu65]. The
basic idea behind the theory can be described as a generalization of the theory
of polynomials in one variable. In the polynomial ring fe[x], where A: is a field,
any ideal J can be generated by a single element, namely the greatest common
divisor of the elements of / . Given any set of generators {/i,... , / s } C k[x]
for J, one can compute (using the Euclidean Algorithm) a single polynomial
d — g c d ( / i , . . . , fs) such that J = ( / i , . . . , fs) = (d). Then a polynomial / € k[x]
is in J if and only if the remainder of the division of / by d is zero. Grobner
bases are the analog of greatest common divisors in the multivariate case in the
following sense. A Grobner basis for an ideal J C fc[#i,... ,x n ] generates I and
a polynomial / G k[x\,... , xn] is in J if and only if the remainder of the division
of / by the polynomials in the Grobner basis is zero (the appropriate concept of
division is a central aspect of the theory).
This abstract characterization of Grobner bases is only one side of the theory.
In fact it falls far short of the true significance of Grobner bases and of the
real contribution of Bruno Buchberger. Indeed, the ideas behind the abstract
characterization of Grobner bases had been around before Buchberger's work.
For example, Macaulay [Mac] used some of these ideas at the beginning of
the century to determine certain invariants of ideals in polynomial rings and
Hironaka [Hi], in 1964, used similar ideas to study power series rings. But the
true significance of Grobner bases is the fact that they can be computed. Bruno
Buchberger's great contribution, and what gave Grobner basis theory the status
as a subject in its own right, is his algorithm for computing these bases.
Our choice of topics is designed to give a broad introduction to the elementary aspects and applications of the subject. As is the case for most topics in
commutative algebra, Grobner basis theory can be presented from a geometric
point of view. We have kept our presentation algebraic except in Sections 1.1
and 2.5. For those interested in a geometric treatment of some of the theory we
recommend the excellent book by D. Cox, J. Little and D. O'Shea [CLOS]. The
reader who is interested in going beyond the contents of this book should use our
list of references as a way to access other sources. We mention in particular the
books by T. Becker and V. Weispfenning [BeWe] and by B. Mishra [Mi] which
contain a lot of material not in this book and have extensive lists of references
on the subject.
Although this book is about computations in algebra, some of the issues which
might be of interest to computer scientists are outside the scope of this book.
For example, implementation of algorithms and their complexity are discussed
only briefly in the book, primarily in Section 3.3. The interested reader should
consult the references.
Wolfgang Grobner was Bruno Buchberger's thesis advisor.
PREFACE
XI
In Chapter 1 we give the basic introduction to the concept of a Grobner basis
and show how to compute it using Buchberger's Algorithm. We are careful to
give motivations for the definition and algorithm by giving the familiar examples
of Gaussian elimination for linear polynomials and the Euclidean Algorithm for
polynomials in one variable. In Chapter 2 we present the basic applications to
algebra and elementary algebraic geometry. We close the chapter with three
specialized applications to algebra, graph theory, and integer programming. In
Chapter 3 we begin by using the concept of syzygy modules to give an improvement of Buchberger's Algorithm. We go on to show how to use Grobner bases to
compute the syzygy module of a set of polynomials (this is solving diophantine
equations over polynomial rings). We then develop the theory of Grobner bases
for finitely generated modules over polynomial rings. With these, we extend
the applications from the previous chapter, give more efficient methods for computing some of the objects from the previous chapter, and conclude by showing
how to compute the Hom functor and free resolutions. In Chapter 4 we develop
the theory of Grobner bases for polynomial rings when the coefficients are now
allowed to be in a general Noetherian ring and we show how to compute these
bases (given certain computability conditions on the coefficient ring). We show
how the theory simplifies when the coefficient ring is a principal ideal domain.
We also give applications to determining whether an ideal is prime and to computing the primary decomposition of ideals in polynomial rings in one variable
over principal ideal domains.
We give an outline of the section dependencies at the end of the Preface.
After Chapter 1 the reader has many options in continuing with the rest of the
book. There are exercises at the end of each section. Many of these exercises
are computational in nature, some doable by hand while others require the use
of a Computer Algebra System. Other exercises extend the theory presented in
the book. A few harder exercises are marked with (*).
This book grew out of a series of lectures presented by the first author at the
National Security Agency during the summer of 1991 and by the second author
at the University of Calabria, Italy, during the summer of 1993.
We would like to thank many of our colleagues and students for their helpful
comments and suggestions. In particular we would like to thank Beth Arnold,
Ann Boyle, Garry Helzer, Karen Horn, Perpetua Kessy, Lyn Miller, Alyson
Reeves, Elizabeth Rutman, Brian Williams, and Eric York. We also want to
thank Sam Rankin, Julie Hawks and the AMS staff for their help in the preparation of the manuscript.
PREFACE
Xll
Chapter 2
Chapter 3
PREFACE
Chapter 4
3.2
4.1
4.2
| 2.4 | | 3.4 |
4.5
4.3
4.6
4.4
xni
References
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[Bu65] B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des
Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. Thesis, Inst. University of Innsbruck, Innsbruck, Austria, 1965.
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[Bu85] B. Buchberger, Grobner bases: An algorithmic method in polynomial
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An Introduction to Computational Algebraic Geometry and Commutative
Algebra, Springer Verlag, Berlin and New York, 1992.
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List of Symbols
N
Z
Zn
Q
M
C
Tn
k
kn
k[x\,...
{fii • • • > fs)
/ = g (mod J)
k[x\,... , xn]/I
/ -f- /
V(S)
V{fi,...
, fa)
I(V)
(S)
gcd
lcm
dim*; (L)
lt(/)
lp(/)
lc(/)
natural numbers
ring of integers
ring of integers modulo n
field of rational numbers
field of real numbers
of complex numbers
set of power products in the variables x\,... , xn
a field
affine space
ring of polynomials in the variables sci,... , x n with
coefficients in the field k
ideal (submodule) generated by / i , . . . , / s
/ congruent to g modulo /
quotient ring of k[x\,... , x n ] by the ideal J
coset of / modulo J
variety in kn defined by the set of polynomials S
variety in kn defined by the polynomials / i , . . . , fs
ideal in k[x\,... , xn] defined by V C kn
ideal generated by the polynomials in the set S
greatest common divisor
least common multiple
dimension of the A;-vector space L
leading term of /
leading power product of /
leading coefficient of /
X <*
<rai
f —> h
f reduces to h modulo G in one step
f —> + h
f reduces to h modulo G
field
,x n ]
. . . Ta«
283
LIST OF SYMBOLS
284
f ^ h
U(S)
S(f,g)
NGU)
VK(S)
k
inj
I: J
im(4>)
ker(<j>)
k[h,...
fs]
k[F]
K\Xi,
. . . , Xn)
k(a)
k(ai,...
,an)
M/N
6 1 , . . . , en
fs ]
fs]
S y z ( / i , . . . fs)
Syz(F)
[fi
[A
'••
-
M/)
Fee
[F\G]
*S
ann(M)
F®G
Hom(M, AT)
(U)
S-XA
AP
Aa
f reduces to h and h = f — Xg
leading term ideal (submodule) defined by S
S-polynomial of / and g
normal form of / with respect to G
variety in Kn denned by S C k[xi,...
,xn]
algebraic closure of the field k
radical of the ideal J
intersection of J and J
ideal quotient of J by J
image of the map 0
kernel of the map (j)
A:-algebra generated by the polynomials / 1 , . . . , fs
fc-algebra generated by the polynomials in the set F
rational function field
field extension of k obtained by adjoining a
field extension of k obtained by adjoining a\,... ,OL
set of column vectors with entries in the ring A
quotient module of M by N
isomorphic to
standard basis for the free module Am
1 x 5 matrix whose entries are polynomials / 1 , . . • 1 Js
m x 5 matrix whose columns are vectors / l 5 . . . ,fs
svzv
g y module of the matrix [ f\ • • • fs ]
syzygy module of the matrix F
leading monomial of the vector /
direct sum of the matrices F and G
concatenation of the matrices F and G
transpose of the matrix S
annihilator of the module M
tensor product of the matrices F and G
set of all A-module homomorphisms between M and N
module generated by the columns of the matrix U
localization of the ring A at the multiplicative set S
localization of the ring A at the prime ideal P
localization of the ring A at the set {1, g, g2, # 3 , . . . }
n
Index
affine
algebra, 84, 180
space, 2
algebra, 232
affine, 84, 180
First Isomorphism Theorem, 80
homomorphism, 79, 92, 232
membership problem, 39
algebraic
closure, 62
element, 97
geometry, 90
algebraically closed field, 62
algorithm, 275
crit2, 130
division
module case, 145
multivariable case, 28
one variable case, 12
ring case, 207
Euclidean, 14
Grobner basis
Buchberger's, field case, 43
Buchberger's, module case, 149
improved Buchberger's, 129
PID case, 250
ring case, 216
ring case using Moller's techniqi
219
primality test, 244
annihilator, 177, 217
ascending chain condition, 6
algorithm for modules, 149
improved algorithm, 129
Theorem, 40, 48
CoCoA, 275
column vector, 113
commutative ring, 1
computation, 275
Computer Algebra System, vii, 275
confluence relation, 37
congruence, 5
Conti, 105
coset, 5
field case
multivariable case, 57
one variable case, 15
module case, 155
ring case, 227
cost function, 105
critl, 128, 152, 258
crit2, 128, 168, 224
deglex, 20
degrevlex, 20
determine, 5, 53
Dickson's Lemma, 23
direct sum of matrices, 174
division
for vectors, 141
multivariable case, 25, 26, 27
one variable case, 11
division algorithm
field case
multivariable case, 28
one variable case, 12
module case, 145
ring case, 207
basis
for A m / M , 155
for k[xi,...
, x n ] / / , 58
in Am, 114
standard in Am, 114
Bayer, 102
Buchberger, viii, 39, 113, 124, 131
algorithm, ix, 43
effective, 53
effective coset representatives, 226
285
286
elimination
Gauss-Jordan, 2
ideal
field case, 70
ring case, 229
module, 156
order, 69, 229
Euclidean Algorithm, 14
exact sequence, 185
short, 184
explicitly given module, 117
Ext functor, 194, 199
Faugere, 68
field
algebraic closure, 62
algebraically closed, 62
First Isomorphism Theorem
algebra case, 80
module case, 117
free module, 114
free resolution, 195
Gauss-Jordan elimination, 2
generating set, 3, 114
finite, 6
Gianni, 68, 237
global dimension, 196
graph, 102
graph of a map, 91
greatest common divisor, 13, 15, 71
Grobner basis
algorithm
Buchberger's, field case, 43
Buchberger's, module case, 148
improved Buchberger's, 129
PID case, 250
ring case, 216
ring case using Moller's technique,
219
for ideal, field case, 32, 34
minimal, 47
reduced, 48
universal, 50
using syzygies, 121
INDEX
for ideal, ring case, 201, 208
minimal, 211
minimal strong, 251, 254, 262
strong, 251
for module, 147
reduced, 150
Hensel's Lemma, 267
Hilbert
Basis Theorem, 5, 6
Nullstellensatz, 62
Horn, 183
homogeneous
ideal, 49
component, 22
polynomial, 22, 38
syzygy, 121, 212
homogenization, 38
homological algebra, 185
homomorphism
algebra, 79, 92, 232
image of, 80
kernel of, 80, 232
module, 116
ideal, 3
field case
elimination, 70
Grobner basis for, 32, 34
homogeneous, 49
intersection, 70, 172, 175
leading term, 32
membership problem, 5, 53
monomial, 23
of relations, 80
quotient, 72
quotient for modules, 157, 177, 182
irreducible, 261
leading term, 32, 206
monomial, 23
P- primary, 261
primary, 260
prime, 61, 237, 259
principal, 13, 246
radical of, 62, 259
287
INDEX
ring case
elimination, 229
Grobner basis for, 201, 208
intersection, 230
leading term, 206
membership problem, 225
of relations, 232
quotient, 217, 231
saturation of, 238
zero-dimensional, 64, 262
image of a homomorphism, 80
implicitization, 91
induction, 277
integer programming, 105
inter-reduced, 131
intersection
of ideals
field case, 70, 172, 175
ring case, 230
of modules, 157, 175, 176, 178
inverse in k[xi,...
, x n ] / 7 , 59, 182
irreducible ideal, 261
isomorphic varieties, 94
isomorphism, module, 116
kernel of a homomorphism, 80, 232
Lakshman, 77
Lazard, 68, 201, 254, 259
leading coefficient
multi-variable case, 21, 202
one variable case, 10
vector case, 143
leading monomial, 143
leading power product, 21, 202
leading term
ideal, 32, 206
module, 147
m u l t i v a r i a t e case, 21, 202
one variable case, 10
vector case, 143
least common multiple
of polynomials, 71
of vectors, 147
lex, 19
linear equations are solvable, 204, 262
local ring, 238
localization, 89
long division, 10
membership problem
algebra, 39
ideal over field, 5, 53
ideal over ring, 225
module, 153
minimal
Grobner basis, field case, 47
Grobner basis, ring case, 211
strong Grobner basis, 251
with respect to F , 205
minimal polynomial, 97
module, 113
elimination, 156
explicitly given, 117
First Isomorphism Theorem, 117
free, 114
Grobner basis for, 147
homomorphism, 116
ideal quotient, 157, 177, 182
intersection, 157, 175, 176, 178
isomorphism, 116
leading term module, 147
membership problem, 153
Noetherian, 116
normal form, 155
quotient, 116
reduction in, 143
S-polynomial, 148
syzygy, 118
Moller, 170, 201, 216, 250
monic polynomial, 13
monoid, 37
monomial
ideal, 23
in modules, 141
leading, 143
Mora, 23, 68
multiplicative set, 238
Noetherian
288
module, 116
ring, 6, 259
normal form
polynomial case
over a field, 57
over a ring, 227
vector case, 155
normal selection strategy, 130
Nullstellensatz, 62
order, 18
degree lexicographic (deglex), 19
degree reverse lexicographic
(degrevlex), 20
elimination, 69
induced by a matrix, 166
lexicographic (lex), 19
position over term (POT), 142
term, 18, 140
term over position (TOP), 142
total, 18, 277
well-ordering, 18, 277
P-primary ideal, 261
parametrized variety, 91
polynomial, 1
elementary symmetric, 25
homogeneous, 22, 38
minimal, 97
monic, 13
normal form, 57, 227
reduced, 27
square free, 75
symmetric, 25, 88
position over term (POT), 142
power product, 1, 18
leading, 21, 202
presentation, 117, 119, 195
primality test, 244
primary
component, 261
decomposition, 261
ideal, 260
prime
component, 261
INDEX
ideal, 61, 237, 259
principal ideal, 13, 246
Principal Ideal Domain (PID), 13, 246
projection map, 90
pullback, 182
quotient
module, 116
ring, 5
radical of an ideal, 62, 259
reduced
Grobner basis
ideal case over fields, 48
module case, 150
polynomial over a field, 27
totally, 227
vector, 144
reduction
linear case, 8
module case, 143
multivariable case over a field, 25, 26,
27
multivariable case over a ring, 203, 205
one variable case, 11
strong, 252, 258
remainder
module case, 144
multivariable case over a field, 27
one variable case over a field, 11
ring
commutative, 1
local, 238
localization, 89
Noetherian, 6, 259
of fractions, 238
quotient, 5
Robbiano, 23, 39
row echelon form, 2
S-basis, 247
S-polynomial
module case, 148
multivariable case over a field, 40, 121,
124
INDEX
PID case, 249
ring case, 213
SAGBI basis, 39
saturated set, 213, 226
saturation of a set, 213
saturation of an ideal, 238
Schreyer, 165
Seidenberg, 78
Shannon, 82, 88
square free polynomial, 75
standard basis, 32
strong
Grobner basis, 251
minimal Grobner basis, 251, 254,
reduction, 252, 258
subalgebra, 39
submodule, 114
Sweedler, 39, 82, 88
symmetric polynomial, 25, 88
syzygy
and Grobner bases, 121
applications of, 171
homogeneous, 121, 212
of a matrix, 161
module of, 161
of vectors over a field, 118
module of, 118, 134
of vectors over a ring, 212
module of, 232, 246
Szekeres, 254
tensor product of matrices, 189
term
module case, 140
order
module case, 140
polynomial case, 18
polynomial case, 1, 18
term over position ( T O P ) , 142
three color problem, 102
total order, 18, 277
totally reduced, 227
Trager, 237
transpose matrix, 176
Traverso, 105
289
universal Grobner basis, 50
variety, 2, 61, 90
isomorphic, 94
parametrized, 91
vector space, 1
well-ordering, 18, 277
Zacharias, 201, 226, 237
Zariski closure, 90
zero divisor, 61
zero-dimensional ideal, 64, 262