Undergraduate Texts in Mathematics
Editors
S. Axler
F.W. Gehring
K.A. Ribet
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Undergraduate Texts in Mathematics
Abbott: Understanding Analysis.
Anglin: Mathematics: A Concise History
and Philosophy.
Readings in Mathematics.
Anglin/Lambek: The Heritage of Thales.
Readings in Mathematics.
Apostol: Introduction to Analytic Number
Theory. Second edition.
Armstrong: Basic Topology.
Armstrong: Groups and Symmetry.
Axler: Linear Algebra Done Right. Second
edition.
Beardon: Limits: A New Approach to Real
Analysis.
Bak/Newman: Complex Analysis. Second
edition.
Banchoff/Wermer: Linear Algebra Through
Geometry. Second edition.
Berberian: A First Course in Real Analysis.
Bix: Conics and Cubics: A Concrete
Introduction to Algebraic Curves.
Brèmaud: An Introduction to Probabilistic
Modeling.
Bressoud: Factorization and Primality Testing.
Bressoud: Second Year Calculus.
Readings in Mathematics.
Brickman: Mathematical Introduction to
Linear Programming and Game Theory.
Browder: Mathematical Analysis: An
Introduction.
Buchmann: Introduction to Cryptography.
Second edition.
Buskes/van Rooij: Topological Spaces: From
Distance to Neighborhood.
Callahan: The Geometry of Spacetime: An
Introduction to Special and General
Relavitity.
Carter/van Brunt: The Lebesgue– Stieltjes
Integral: A Practical Introduction.
Cederberg: A Course in Modern
Geometries. Second edition.
Chambert-Loir: A Field Guide to Algebra
Childs: A Concrete Introduction to Higher
Algebra. Second edition.
Chung/AitSahlia: Elementary Probability
Theory: With Stochastic Processes and an
Introduction to Mathematical Finance.
Fourth edition.
Cox/Little/O’Shea: Ideals, Varieties, and
Algorithms. Third edition. (2007)
Croom: Basic Concepts of Algebraic
Topology.
Cull/Flahive/Robson: Difference Equations.
From Rabbits to Chaos.
Curtis: Linear Algebra: An Introductory
Approach. Fourth edition.
Daepp/Gorkin: Reading, Writing, and
Proving: A Closer Look at Mathematics.
Devlin: The Joy of Sets: Fundamentals
of Contemporary Set Theory. Second
edition.
Dixmier: General Topology.
Driver: Why Math?
Ebbinghaus/Flum/Thomas: Mathematical
Logic. Second edition.
Edgar: Measure, Topology, and Fractal
Geometry.
Elaydi: An Introduction to Difference
Equations. Third edition.
Erdõs/Surányi: Topics in the Theory of
Numbers.
Estep: Practical Analysis on One Variable.
Exner: An Accompaniment to Higher
Mathematics.
Exner: Inside Calculus.
Fine/Rosenberger: The Fundamental Theory
of Algebra.
Fischer: Intermediate Real Analysis.
Flanigan/Kazdan: Calculus Two: Linear and
Nonlinear Functions. Second edition.
Fleming: Functions of Several Variables.
Second edition.
Foulds: Combinatorial Optimization for
Undergraduates.
Foulds: Optimization Techniques: An
Introduction.
Franklin: Methods of Mathematical
Economics.
Frazier: An Introduction to Wavelets
Through Linear Algebra.
Gamelin: Complex Analysis.
Ghorpade/Limaye: A Course in Calculus and
Real Analysis.
Gordon: Discrete Probability.
Hairer/Wanner: Analysis by Its History.
Readings in Mathematics.
Halmos: Finite-Dimensional Vector Spaces.
Second edition.
Halmos: Naive Set Theory.
Hämmerlin/Hoffmann: Numerical
Mathematics.
Readings in Mathematics.
Harris/Hirst/Mossinghoff: Combinatorics
and Graph Theory.
Hartshorne: Geometry: Euclid and
Beyond.
Hijab: Introduction to Calculus and
Classical Analysis.
Hilton/Holton/Pedersen: Mathematical
Reflections: In a Room with Many
Mirrors.
Hilton/Holton/Pedersen: Mathematical
Vistas: From a Room with Many
Windows.
(continued after index)
www.pdfgrip.com
David Cox
John Little
Donal O’Shea
Ideals, Varieties, and
Algorithms
An Introduction to Computational Algebraic
Geometry and Commutative Algebra
Third Edition
www.pdfgrip.com
David Cox
Department of Mathematics
and Computer Science
Amherst College
Amherst, MA 01002-5000
USA
Editorial Board
S. Axler
Mathematics Department
San Francisco State
University
San Francisco, CA 94132
USA
John Little
Department of Mathematics
College of the Holy Cross
Worcester, MA 01610-2395
USA
Donal O’Shea
Department of Mathematics
and Statistics
Mount Holyoke College
South Hadley, MA 01075-1493
USA
F.W. Gehring
Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109
USA
K.A. Ribet
Department of Mathematics
University of California
at Berkeley
Berkeley, CA 94720-3840
USA
Mathematics Subject Classification (2000): 14-01, 13-01, 13Pxx
Library of Congress Control Number: 2006930875
ISBN-10: 0-387-35650-9
ISBN-13: 978-0-387-35650-1
e-ISBN-10: 0-387-35651-7
e-ISBN-13: 978-0-387-35651-8
Printed on acid-free paper.
© 2007, 1997, 1992 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic
adaptation, computer software, or by similar or dissimilar methodology now known or hereafter
developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or
not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 1
springer.com
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To Elaine,
for her love and support.
D.A.C.
To my mother and the memory of my father.
J.B.L.
To Mary and my children.
D.O’S.
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Preface to the First Edition
We wrote this book to introduce undergraduates to some interesting ideas in algebraic
geometry and commutative algebra. Until recently, these topics involved a lot of abstract
mathematics and were only taught in graduate school. But in the 1960s, Buchberger
and Hironaka discovered new algorithms for manipulating systems of polynomial equations. Fueled by the development of computers fast enough to run these algorithms,
the last two decades have seen a minor revolution in commutative algebra. The ability
to compute efficiently with polynomial equations has made it possible to investigate
complicated examples that would be impossible to do by hand, and has changed the
practice of much research in algebraic geometry. This has also enhanced the importance of the subject for computer scientists and engineers, who have begun to use these
techniques in a whole range of problems.
It is our belief that the growing importance of these computational techniques warrants their introduction into the undergraduate (and graduate) mathematics curriculum.
Many undergraduates enjoy the concrete, almost nineteenth-century, flavor that a computational emphasis brings to the subject. At the same time, one can do some substantial mathematics, including the Hilbert Basis Theorem, Elimination Theory, and the
Nullstellensatz.
The mathematical prerequisites of the book are modest: the students should have had
a course in linear algebra and a course where they learned how to do proofs. Examples
of the latter sort of course include discrete math and abstract algebra. It is important to
note that abstract algebra is not a prerequisite. On the other hand, if all of the students
have had abstract algebra, then certain parts of the course will go much more quickly.
The book assumes that the students will have access to a computer algebra system.
Appendix C describes the features of AXIOM, Maple, Mathematica, and REDUCE that
are most relevant to the text. We do not assume any prior experience with a computer.
However, many of the algorithms in the book are described in pseudocode, which may
be unfamiliar to students with no background in programming. Appendix B contains a
careful description of the pseudocode that we use in the text.
In writing the book, we tried to structure the material so that the book could be used
in a variety of courses, and at a variety of different levels. For instance, the book could
serve as a basis of a second course in undergraduate abstract algebra, but we think that
it just as easily could provide a credible alternative to the first course. Although the
vii
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Preface to the First Edition
book is aimed primarily at undergraduates, it could also be used in various graduate
courses, with some supplements. In particular, beginning graduate courses in algebraic
geometry or computational algebra may find the text useful. We hope, of course, that
mathematicians and colleagues in other disciplines will enjoy reading the book as much
as we enjoyed writing it.
The first four chapters form the core of the book. It should be possible to cover them
in a 14-week semester, and there may be some time left over at the end to explore other
parts of the text. The following chart explains the logical dependence of the chapters:
1
2
3
4
8
6
5
7
9
See the table of contents for a description of what is covered in each chapter. As the
chart indicates, there are a variety of ways to proceed after covering the first four
chapters. Also, a two-semester course could be designed that covers the entire book.
For instructors interested in having their students do an independent project, we have
included a list of possible topics in Appendix D.
It is a pleasure to thank the New England Consortium for Undergraduate Science
Education (and its parent organization, the Pew Charitable Trusts) for providing the
major funding for this work. The project would have been impossible without their
support. Various aspects of our work were also aided by grants from IBM and the Sloan
Foundation, the Alexander von Humboldt Foundation, the Department of Education’s
FIPSE program, the Howard Hughes Foundation, and the National Science Foundation.
We are grateful for their help.
We also wish to thank colleagues and students at Amherst College, George Mason
University, Holy Cross College, Massachusetts Institute of Technology, Mount Holyoke
College, Smith College, and the University of Massachusetts who participated in courses based on early versions of the manuscript. Their feedback improved the book considerably. Many other colleagues have contributed suggestions, and we thank you all.
Corrections, comments and suggestions for improvement are welcome!
David Cox
John Little
Donal O’ Shea
November 1991
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Preface to the Second Edition
In preparing a new edition of Ideals, Varieties, and Algorithms, our goal was to
correct some of the omissions of the first edition while maintaining the readability and accessibility of the original. The major changes in the second edition are as
follows:
r Chapter 2: A better acknowledgement of Buchberger’s contributions and an improved
proof of the Buchberger Criterion in §6.
r Chapter 5: An improved bound on the number of solutions in §3 and a new §6 which
completes the proof of the Closure Theorem begun in Chapter 3.
r Chapter 8: A complete proof of the Projection Extension Theorem in §5 and a new
§7 which contains a proof of Bezout’s Theorem.
r Appendix C: a new section on AXIOM and an update on what we say about Maple,
Mathematica, and REDUCE.
Finally, we fixed some typographical errors, improved and clarified notation, and updated the bibliography by adding many new references.
We also want to take this opportunity to acknowledge our debt to the many people
who influenced us and helped us in the course of this project. In particular, we would
like to thank:
r David Bayer and Monique Lejeune-Jalabert, whose thesis BAYER (1982) and notes
LEJEUNE-JALABERT (1985) first acquainted us with this wonderful subject.
r Frances Kirwan, whose book KIRWAN (1992) convinced us to include Bezout’s
Theorem in Chapter 8.
r Steven Kleiman, who showed us how to prove the Closure Theorem in full generality.
His proof appears in Chapter 5.
r Michael Singer, who suggested improvements in Chapter 5, including the new Proposition 8 of §3.
r Bernd Sturmfels, whose book STURMFELS (1993) was the inspiration for
Chapter 7.
There are also many individuals who found numerous typographical errors and gave
us feedback on various aspects of the book. We are grateful to you all!
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Preface to the Second Edition
As with the first edition, we welcome comments and suggestions, and we pay $1 for
every new typographical error. For a list of errors and other information relevant to the
book, see our web site />David Cox
John Little
Donal O’ Shea
October 1996
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Preface to the Third Edition
The new features of the third edition of Ideals, Varieties, and Algorithms are as follows:
r A significantly shorter proof of the Extension Theorem is presented in §6 of Chapter 3.
We are grateful to A. H. M. Levelt for bringing this argument to our attention.
r A major update of the section on Maple appears in Appendix C. We also give
updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica, and
SINGULAR.
r Changes have been made on over 200 pages to enhance clarity and correctness.
We are also grateful to the many individuals who reported typographical errors and
gave us feedback on the earlier editions. Thank you all!
As with the first and second editions, we welcome comments and suggestions, and
we pay $1 for every new typographical error.
David Cox
John Little
Donal O’ Shea
November, 2006
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Contents
Preface to the First Edition
vii
Preface to the Second Edition
ix
Preface to the Third Edition
xi
1. Geometry, Algebra, and Algorithms
1
§1. Polynomials and Affine Space . .
§2. Affine Varieties . . . . . . . .
§3. Parametrizations of Affine Varieties
§4. Ideals . . . . . . . . . . . .
§5. Polynomials of One Variable . . .
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2. Groebner Bases
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§1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
§2. Orderings on the Monomials in k[x1 , . . . , xn ] . . . . . . . . . .
§3. A Division Algorithm in k[x1 , . . . , xn ] . . . . . . . . . . . . .
§4. Monomial Ideals and Dickson’s Lemma . . . . . . . . . . . . .
§5. The Hilbert Basis Theorem and Groebner Bases . . . . . . . . .
§6. Properties of Groebner Bases . . . . . . . . . . . . . . . . .
§7. Buchberger’s Algorithm . . . . . . . . . . . . . . . . . . .
§8. First Applications of Groebner Bases . . . . . . . . . . . . . .
§9. (Optional) Improvements on Buchberger’s Algorithm . . . . . . .
3. Elimination Theory
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115
§1. The Elimination and Extension Theorems .
§2. The Geometry of Elimination . . . . . .
§3. Implicitization . . . . . . . . . . . .
§4. Singular Points and Envelopes . . . . .
§5. Unique Factorization and Resultants . . .
§6. Resultants and the Extension Theorem . .
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115
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Contents
4. The Algebra–Geometry Dictionary
169
§1. Hilbert’s Nullstellensatz . . . . . . . . . . . .
§2. Radical Ideals and the Ideal–Variety Correspondence
§3. Sums, Products, and Intersections of Ideals . . . .
§4. Zariski Closure and Quotients of Ideals . . . . . .
§5. Irreducible Varieties and Prime Ideals . . . . . .
§6. Decomposition of a Variety into Irreducibles . . . .
§7. (Optional) Primary Decomposition of Ideals . . . .
§8. Summary . . . . . . . . . . . . . . . . . .
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5. Polynomial and Rational Functions on a Variety
§1. Polynomial Mappings . . . . . . . . . .
§2. Quotients of Polynomial Rings . . . . . .
§3. Algorithmic Computations in k[x1 , . . . , xn ]/I
§4. The Coordinate Ring of an Affine Variety . .
§5. Rational Functions on a Variety . . . . . .
§6. (Optional) Proof of the Closure Theorem . .
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6. Robotics and Automatic Geometric Theorem Proving
§1. Geometric Description of Robots . . . . . . . .
§2. The Forward Kinematic Problem . . . . . . . .
§3. The Inverse Kinematic Problem and Motion Planning
§4. Automatic Geometric Theorem Proving . . . . . .
§5. Wu’s Method . . . . . . . . . . . . . . . .
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7. Invariant Theory of Finite Groups
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317
§1. Symmetric Polynomials . . . . . . . . . . . . .
§2. Finite Matrix Groups and Rings of Invariants . . . . .
§3. Generators for the Ring of Invariants . . . . . . . .
§4. Relations Among Generators and the Geometry of Orbits
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8. Projective Algebraic Geometry
§1. The Projective Plane . . . . . . . . . .
§2. Projective Space and Projective Varieties . .
§3. The Projective Algebra–Geometry Dictionary
§4. The Projective Closure of an Affine Variety .
§5. Projective Elimination Theory . . . . . .
§6. The Geometry of Quadric Hypersurfaces . .
§7. Bezout’s Theorem . . . . . . . . . . .
169
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357
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9. The Dimension of a Variety
357
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439
§1. The Variety of a Monomial Ideal . . . . . . . . . . . . . . .
§2. The Complement of a Monomial Ideal . . . . . . . . . . . . .
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Contents
§3. The Hilbert Function and the Dimension of a Variety
§4. Elementary Properties of Dimension . . . . . . .
§5. Dimension and Algebraic Independence . . . . .
§6. Dimension and Nonsingularity . . . . . . . . .
§7. The Tangent Cone . . . . . . . . . . . . . .
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Appendix A. Some Concepts from Algebra
Appendix B. Pseudocode
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Appendix C. Computer Algebra Systems
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509
510
511
513
§1. Inputs, Outputs, Variables, and Constants
§2. Assignment Statements . . . . . . .
§3. Looping Structures . . . . . . . . .
§4. Branching Structures . . . . . . . .
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456
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477
484
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509
§1. Fields and Rings . . . . . . . . . . . . . . . . . . . . . .
§2. Groups . . . . . . . . . . . . . . . . . . . . . . . . . .
§3. Determinants . . . . . . . . . . . . . . . . . . . . . . .
§1. AXIOM . . .
§2. Maple . . . .
§3. Mathematica .
§4. REDUCE . . .
§5. Other Systems .
xv
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513
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Appendix D. Independent Projects
§1. General Comments . . . . . . . . . . . . . . . . . . . . .
§2. Suggested Projects . . . . . . . . . . . . . . . . . . . . .
517
520
522
524
528
530
530
530
References
535
Index
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1
Geometry, Algebra, and Algorithms
This chapter will introduce some of the basic themes of the book. The geometry we
are interested in concerns affine varieties, which are curves and surfaces (and higher
dimensional objects) defined by polynomial equations. To understand affine varieties,
we will need some algebra, and in particular, we will need to study ideals in the
polynomial ring k[x1 , . . . , xn ]. Finally, we will discuss polynomials in one variable to
illustrate the role played by algorithms.
§1 Polynomials and Affine Space
To link algebra and geometry, we will study polynomials over a field. We all know what
polynomials are, but the term field may be unfamiliar. The basic intuition is that a field
is a set where one can define addition, subtraction, multiplication, and division with the
usual properties. Standard examples are the real numbers and the complex numbers
, whereas the integers are not a field since division fails (3 and 2 are integers, but
their quotient 3/2 is not). A formal definition of field may be found in Appendix A.
One reason that fields are important is that linear algebra works over any field. Thus,
even if your linear algebra course restricted the scalars to lie in or , most of the
theorems and techniques you learned apply to an arbitrary field k. In this book, we will
employ different fields for different purposes. The most commonly used fields will be:
r The rational numbers : the field for most of our computer examples.
r The real numbers : the field for drawing pictures of curves and surfaces.
r The complex numbers : the field for proving many of our theorems.
On occasion, we will encounter other fields, such as fields of rational functions (which
will be defined later). There is also a very interesting theory of finite fields—see the
exercises for one of the simpler examples.
We can now define polynomials. The reader certainly is familiar with polynomials in
one and two variables, but we will need to discuss polynomials in n variables x1 , . . . , xn
with coefficients in an arbitrary field k. We start by defining monomials.
Definition 1. A monomial in x1 , . . . , xn is a product of the form
x1α1 · x2α2 · · · xnαn ,
1
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where all of the exponents α1 , . . . , αn are nonnegative integers. The total degree of
this monomial is the sum α1 + · · · + αn .
We can simplify the notation for monomials as follows: let α = (α1 , . . . , αn ) be an
n-tuple of nonnegative integers. Then we set
x α = x1α1 · x2α2 · · · xnαn .
When α = (0, . . . , 0), note that x α = 1. We also let |α| = α1 + · · · + αn denote the
total degree of the monomial x α .
Definition 2. A polynomial f in x1 , . . . , xn with coefficients in k is a finite linear
combination (with coefficients in k) of monomials. We will write a polynomial f in the
form
f =
α
aα x α ,
aα ∈ k,
where the sum is over a finite number of n-tuples α = (α1 , . . . , αn ). The set of all
polynomials in x1 , . . . , xn with coefficients in k is denoted k[x1 , . . . , xn ].
When dealing with polynomials in a small number of variables, we will usually
dispense with subscripts. Thus, polynomials in one, two, and three variables lie in
k[x], k[x, y] and k[x, y, z], respectively. For example,
f = 2x 3 y 2 z +
3 3 3
y z − 3x yz + y 2
2
is a polynomial in [x, y, z]. We will usually use the letters f, g, h, p, q, r to refer to
polynomials.
We will use the following terminology in dealing with polynomials.
Definition 3. Let f = α aα x α be a polynomial in k[x1 , . . . , xn ].
(i) We call aα the coefficient of the monomial x α .
(ii) If aα = 0, then we call aα x α a term of f.
(iii) The total degree of f, denoted deg( f ), is the maximum |α| such that the coefficient
aα is nonzero.
As an example, the polynomial f = 2x 3 y 2 z + 32 y 3 z 3 − 3x yz + y 2 given above has
four terms and total degree six. Note that there are two terms of maximal total degree,
which is something that cannot happen for polynomials of one variable. In Chapter 2,
we will study how to order the terms of a polynomial.
The sum and product of two polynomials is again a polynomial. We say that a
polynomial f divides a polynomial g provided that g = f h for some h ∈ k[x1 , . . . , xn ].
One can show that, under addition and multiplication, k[x1 , . . . , xn ] satisfies all of the
field axioms except for the existence of multiplicative inverses (because, for example,
1/x1 is not a polynomial). Such a mathematical structure is called a commutative ring
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3
(see Appendix A for the full definition), and for this reason we will refer to k[x1 , . . . , xn ]
as a polynomial ring.
The next topic to consider is affine space.
Definition 4. Given a field k and a positive integer n, we define the n-dimensional
affine space over k to be the set
k n = {(a1 , . . . , an ) : a1 , . . . , an ∈ k}.
For an example of affine space, consider the case k = . Here we get the familiar
space n from calculus and linear algebra. In general, we call k 1 = k the affine line
and k 2 the affine plane.
Let us next see how polynomials relate to affine space. The key idea is that a polynomial f = α aα x α ∈ k[x1 , . . . , xn ] gives a function
f : kn → k
defined as follows: given (a1 , . . . , an ) ∈ k n , replace every xi by ai in the expression
for f . Since all of the coefficients also lie in k, this operation gives an element
f (a1 , . . . , an ) ∈ k. The ability to regard a polynomial as a function is what makes
it possible to link algebra and geometry.
This dual nature of polynomials has some unexpected consequences. For example,
the question “is f = 0?” now has two potential meanings: is f the zero polynomial?,
which means that all of its coefficients aα are zero, or is f the zero function?, which
means that f (a1 , . . . , an ) = 0 for all (a1 , . . . , an ) ∈ k n . The surprising fact is that these
two statements are not equivalent in general. For an example of how they can differ,
consider the set consisting of the two elements 0 and 1. In the exercises, we will see
that this can be made into a field where 1 + 1 = 0. This field is usually called 2 . Now
consider the polynomial x 2 − x = x(x − 1) ∈ 2 [x]. Since this polynomial vanishes
at 0 and 1, we have found a nonzero polynomial which gives the zero function on the
affine space 12 . Other examples will be discussed in the exercises.
However, as long as k is infinite, there is no problem.
Proposition 5. Let k be an infinite field, and let f ∈ k[x1 , . . . , xn ]. Then f = 0 in
k[x1 , . . . , xn ] if and only if f : k n → k is the zero function.
Proof. One direction of the proof is obvious since the zero polynomial clearly gives
the zero function. To prove the converse, we need to show that if f (a1 , . . . , an ) = 0
for all (a1 , . . . , an ) ∈ k n , then f is the zero polynomial. We will use induction on the
number of variables n.
When n = 1, it is well known that a nonzero polynomial in k[x] of degree m has at
most m distinct roots (we will prove this fact in Corollary 3 of §5). For our particular
f ∈ k[x], we are assuming f (a) = 0 for all a ∈ k. Since k is infinite, this means that
f has infinitely many roots, and, hence, f must be the zero polynomial.
Now assume that the converse is true for n − 1, and let f ∈ k[x1 , . . . , xn ] be a
polynomial that vanishes at all points of k n . By collecting the various powers of xn , we
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can write f in the form
N
f =
gi (x1 , . . . , xn−1 )xni ,
i=0
where gi ∈ k[x1 , . . . , xn−1 ]. We will show that each gi is the zero polynomial in n − 1
variables, which will force f to be the zero polynomial in k[x1 , . . . , xn ].
If we fix (a1 , . . . , an−1 ) ∈ k n−1 , we get the polynomial f (a1 , . . . , an−1 , xn ) ∈ k[xn ].
By our hypothesis on f , this vanishes for every an ∈ k. It follows from the case n = 1
that f (a1 , . . . , an−1 , xn ) is the zero polynomial in k[xn ]. Using the above formula for
f , we see that the coefficients of f (a1 , . . . , an−1 , xn ) are gi (a1 , . . . , an−1 ), and thus,
gi (a1 , . . . , an−1 ) = 0 for all i. Since (a1 , . . . , an−1 ) was arbitrarily chosen in k n−1 , it
follows that each gi ∈ k[x1 , . . . , xn−1 ] gives the zero function on k n−1 . Our inductive
assumption then implies that each gi is the zero polynomial in k[x1 , . . . , xn−1 ]. This
forces f to be the zero polynomial in k[x1 , . . . , xn ] and completes the proof of the
proposition.
Note that in the statement of Proposition 5, the assertion “ f = 0 in k[x1 , . . . , xn ]”
means that f is the zero polynomial, i.e., that every coefficient of f is zero. Thus, we
use the same symbol “0” to stand for the zero element of k and the zero polynomial in
k[x1 , . . . , xn ]. The context will make clear which one we mean.
As a corollary, we see that two polynomials over an infinite field are equal precisely
when they give the same function on affine space.
Corollary 6. Let k be an infinite field, and let f, g ∈ k[x1 , . . . , xn ]. Then f = g in
k[x1 , . . . , xn ] if and only if f : k n → k and g : k n → k are the same function.
Proof. To prove the nontrivial direction, suppose that f, g ∈ k[x1 , . . . , xn ] give the
same function on k n . By hypothesis, the polynomial f − g vanishes at all points of k n .
Proposition 5 then implies that f − g is the zero polynomial. This proves that f = g
in k[x1 , . . . , xn ].
Finally, we need to record a special property of polynomials over the field of complex
numbers .
Theorem 7. Every nonconstant polynomial f ∈
[x] has a root in .
Proof. This is the Fundamental Theorem of Algebra, and proofs can be found in most
introductory texts on complex analysis (although many other proofs are known).
We say that a field k is algebraically closed if every nonconstant polynomial in k[x]
has a root in k. Thus is not algebraically closed (what are the roots of x 2 + 1?),
whereas the above theorem asserts that is algebraically closed. In Chapter 4 we will
prove a powerful generalization of Theorem 7 called the Hilbert Nullstellensatz.
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EXERCISES FOR §1
1. Let 2 = {0, 1}, and define addition and multiplication by 0 + 0 = 1 + 1 = 0, 0 + 1 =
1 + 0 = 1, 0 · 0 = 0 · 1 = 1 · 0 = 0 and 1 · 1 = 1. Explain why 2 is a field. (You need not
check the associative and distributive properties, but you should verify the existence of identities and inverses, both additive and multiplicative.)
2. Let 2 be the field from Exercise 1.
a. Consider the polynomial g(x, y) = x 2 y + y 2 x ∈ 2 [x, y]. Show that g(x, y) = 0 for every (x, y) ∈ 22 , and explain why this does not contradict Proposition 5.
b. Find a nonzero polynomial in 2 [x, y, z] which vanishes at every point of 32 . Try to find
one involving all three variables.
c. Find a nonzero polynomial in 2 [x1 , . . . , xn ] which vanishes at every point of n2 . Can
you find one in which all of x1 , . . . , xn appear?
3. (Requires abstract algebra). Let p be a prime number. The ring of integers modulo p is a field
with p elements, which we will denote p .
a. Explain why p − {0} is a group under multiplication.
b. Use Lagrange’s Theorem to show that a p−1 = 1 for all a ∈ p − {0}.
c. Prove that a p = a for all a ∈ p . Hint: Treat the cases a = 0 and a = 0 separately.
d. Find a nonzero polynomial in p [x] which vanishes at every point of p . Hint: Use
part (c).
4. (Requires abstract algebra.) Let F be a finite field with q elements. Adapt the argument of
Exercise 3 to prove that x q − x is a nonzero polynomial in F[x] which vanishes at every
point of F. This shows that Proposition 5 fails for all finite fields.
5. In the proof of Proposition 5, we took f ∈ k[x1 , . . . , xn ] and wrote it as a polynomial in xn
with coefficients in k[x1 , . . . , xn−1 ]. To see what this looks like in a specific case, consider
the polynomial
f (x, y, z) = x 5 y 2 z − x 4 y 3 + y 5 + x 2 z − y 3 z + x y + 2x − 5z + 3.
a. Write f as a polynomial in x with coefficients in k[y, z].
b. Write f as a polynomial in y with coefficients in k[x, z].
c. Write f as a polynomial in z with coefficients in k[x, y].
6. Inside of n , we have the subset n , which consists of all points with integer coordinates.
a. Prove that if f ∈ [x1 , . . . , xn ] vanishes at every point of n , then f is the zero polynomial. Hint: Adapt the proof of Proposition 5.
b. Let f ∈ [x1 , . . . , xn ], and let M be the largest power of any variable that appears in f .
Let nM+1 be the set of points of n , all coordinates of which lie between 1 and M + 1.
Prove that if f vanishes at all points of nM+1 , then f is the zero polynomial.
§2 Affine Varieties
We can now define the basic geometric object of the book.
Definition 1. Let k be a field, and let f 1 , . . . , f s be polynomials in k[x1 , . . . , xn ]. Then
we set
V( f 1 , . . . , f s ) = {(a1 , . . . , an ) ∈ k n : f i (a1 , . . . , an ) = 0 for all 1 ≤ i ≤ s}.
We call V( f 1 , . . . , f s ) the affine variety defined by f 1 , . . . , f s .
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Thus, an affine variety V( f 1 , . . . , f s ) ⊂ k n is the set of all solutions of the system of
equations f 1 (x1 , . . . , xn ) = · · · = f s (x1 , . . . , xn ) = 0. We will use the letters V, W, etc.
to denote affine varieties. The main purpose of this section is to introduce the reader to
lots of examples, some new and some familiar. We will use k = so that we can draw
pictures.
We begin in the plane 2 with the variety V(x 2 + y 2 − 1), which is the circle of
radius 1 centered at the origin:
y
1
1
x
The conic sections studied in analytic geometry (circles, ellipses, parabolas, and hyperbolas) are affine varieties. Likewise, graphs of polynomial functions are affine varieties
[the graph of y = f (x) is V(y − f (x))]. Although not as obvious, graphs of rational
3
functions are also affine varieties. For example, consider the graph of y = x x−1 :
30 y
20
10
−4
−2
2
−10
−20
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It is easy to check that this is the affine variety V(x y − x 3 + 1).
Next, let us look in the 3-dimensional space 3 . A nice affine variety is given by
paraboloid of revolution V(z − x 2 − y), which is obtained by rotating the parabola
z = x 2 about the z-axis (you can check this using polar coordinates). This gives us the
picture:
z
y
x
You may also be familiar with the cone V(z 2 − x 2 − y 2 ):
z
y
x
A much more complicated surface is given by V(x 2 − y 2 z 2 + z 3 ):
z
x
y
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In these last two examples, the surfaces are not smooth everywhere: the cone has a
sharp point at the origin, and the last example intersects itself along the whole y-axis.
These are examples of singular points, which will be studied later in the book.
An interesting example of a curve in 3 is the twisted cubic, which is the variety
V(y − x 2 , z − x 3 ). For simplicity, we will confine ourselves to the portion that lies in
the first octant. To begin, we draw the surfaces y = x 2 and z = x 3 separately:
z
z
x
y
x
y
O
O
y = x2
z = x3
Then their intersection gives the twisted cubic:
z
x
y
O
The Twisted Cubic
Notice that when we had one equation in 2 , we got a curve, which is a 1-dimensional
object. A similar situation happens in 3 : one equation in 3 usually gives a surface,
which has dimension 2. Again, dimension drops by one. But now consider the twisted
cubic: here, two equations in 3 give a curve, so that dimension drops by two. Since
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each equation imposes an extra constraint, intuition suggests that each equation drops
the dimension by one. Thus, if we started in 4 , one would hope that an affine variety
defined by two equations would be a surface. Unfortunately, the notion of dimension is
more subtle than indicated by the above examples. To illustrate this, consider the variety
V(x z, yz). One can easily check that the equations x z = yz = 0 define the union of
the (x, y)-plane and the z-axis:
z
y
x
Hence, this variety consists of two pieces which have different dimensions, and one of
the pieces (the plane) has the “wrong” dimension according to the above intuition.
We next give some examples of varieties in higher dimensions. A familiar case comes
from linear algebra. Namely, fix a field k, and consider a system of m linear equations
in n unknowns x1 , . . . , xn with coefficients in k:
(1)
a11 x1 + · · · + a1n xn = b1 ,
..
.
am1 x1 + · · · + amn xn = bm .
The solutions of these equations form an affine variety in k n , which we will call a
linear variety. Thus, lines and planes are linear varieties, and there are examples of
arbitrarily large dimension. In linear algebra, you learned the method of row reduction
(also called Gaussian elimination), which gives an algorithm for finding all solutions
of such a system of equations. In Chapter 2, we will study a generalization of this
algorithm which applies to systems of polynomial equations.
Linear varieties relate nicely to our discussion of dimension. Namely, if V ⊂ k n is
the linear variety defined by (1), then V need not have dimension n − m even though
V is defined by m equations. In fact, when V is nonempty, linear algebra tells us that V
has dimension n − r , where r is the rank of the matrix (ai j ). So for linear varieties, the
dimension is determined by the number of independent equations. This intuition applies
to more general affine varieties, except that the notion of “independent” is more subtle.
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Some complicated examples in higher dimensions come from calculus. Suppose, for
example, that we wanted to find the minimum and maximum values of f (x, y, z) =
x 3 + 2x yz − z 2 subject to the constraint g(x, y, z) = x 2 + y 2 + z 2 = 1. The method
of Lagrange multipliers states that ∇ f = λ∇g at a local minimum or maximum [recall
that the gradient of f is the vector of partial derivatives ∇ f = ( f x , f y , f z )]. This gives
us the following system of four equations in four unknowns, x, y, z, λ, to solve:
(2)
3x 2 + 2yz = 2xλ,
2x z = 2yλ,
2x y − 2z = 2zλ,
x 2 + y 2 + z 2 = 1.
These equations define an affine variety in 4 , and our intuition concerning dimension
leads us to hope it consists of finitely many points (which have dimension 0) since it is
defined by four equations. Students often find Lagrange multipliers difficult because
the equations are so hard to solve. The algorithms of Chapter 2 will provide a powerful
tool for attacking such problems. In particular, we will find all solutions of the above
equations.
We should also mention that affine varieties can be the empty set. For example, when
k = , it is obvious that V(x 2 + y 2 + 1) = ∅ since x 2 + y 2 = −1 has no real solutions
(although there are solutions when k = ). Another example is V(x y, x y − 1), which
is empty no matter what the field is, for a given x and y cannot satisfy both x y = 0 and
x y = 1. In Chapter 4 we will study a method for determining when an affine variety
over is nonempty.
To give an idea of some of the applications of affine varieties, let us consider a simple
example from robotics. Suppose we have a robot arm in the plane consisting of two
linked rods of lengths 1 and 2, with the longer rod anchored at the origin:
(x,y)
(z,w)
The “state” of the arm is completely described by the coordinates (x, y) and (z, w)
indicated in the figure. Thus the state can be regarded as a 4-tuple (x, y, z, w) ∈ 4 .
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However, not all 4-tuples can occur as states of the arm. In fact, it is easy to see that
the subset of possible states is the affine variety in 4 defined by the equations
x 2 + y 2 = 4,
(x − z)2 + (y − w)2 = 1.
Notice how even larger dimensions enter quite easily: if we were to consider the same
arm in 3-dimensional space, then the variety of states would be defined by two equations
in 6 . The techniques to be developed in this book have some important applications
to the theory of robotics.
So far, all of our drawings have been over . Later in the book, we will consider
varieties over . Here, it is more difficult (but not impossible) to get a geometric idea
of what such a variety looks like.
Finally, let us record some basic properties of affine varieties.
Lemma 2. If V, W ⊂ k n are affine varieties, then so are V ∪ W and V ∩ W .
Proof. Suppose that V = V( f 1 , . . . , f s ) and W = V(g1 , . . . , gt ). Then we claim that
V ∩ W = V( f 1 , . . . , f s , g1 , . . . , gt ),
V ∪ W = V( f i g j : 1 ≤ i ≤ s, 1 ≤ j ≤ t).
The first equality is trivial to prove: being in V ∩ W means that both f 1 , . . . , f s and
g1 , . . . , gt vanish, which is the same as f 1 , . . . , f s , g1 , . . . , gt vanishing.
The second equality takes a little more work. If (a1 , . . . , an ) ∈ V , then all of the f i ’s
vanish at this point, which implies that all of the f i g j ’s also vanish at (a1 , . . . , an ). Thus,
V ⊂ V( f i g j ), and W ⊂ V( f i g j ) follows similarly. This proves that V ∪ W ⊂ V( f i g j ).
Going the other way, choose (a1 , . . . , an ) ∈ V( f i g j ). If this lies in V , then we are done,
and if not, then f i0 (a1 , . . . , an ) = 0 for some i 0 . Since f i0 g j vanishes at (a1 , . . . , an )
for all j, the g j ’s must vanish at this point, proving that (a1 , . . . , an ) ∈ W . This shows
that V( f i g j ) ⊂ V ∪ W .
This lemma implies that finite intersections and unions of affine varieties are again
affine varieties. It turns out that we have already seen examples of unions and intersections. Concerning unions, consider the union of the (x, y)-plane and the z-axis in
affine 3-space. By the above formula, we have
V(z) ∪ V(x, y) = V(zx, zy).
This, of course, is one of the examples discussed earlier in the section. As for intersections, notice that the twisted cubic was given as the intersection of two surfaces.
The examples given in this section lead to some interesting questions concerning
affine varieties. Suppose that we have f 1 , . . . , f s ∈ k[x1 , . . . , xn ]. Then:
r (Consistency) Can we determine if V( f 1 , . . . , f s ) = ∅, i.e., do the equations f 1 =
· · · = f s = 0 have a common solution?
r (Finiteness) Can we determine if V( f 1 , . . . , f s ) is finite, and if so, can we find all of
the solutions explicitly?
r (Dimension) Can we determine the “dimension” of V( f 1 , . . . , f s )?
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