http://dx.doi.org/10.1090/gsm/003 Selected Titles in This Series

13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 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'JJ 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. The book contains not only the theory over fields, but also, the theory in modules and over rings.

Library of Congress Cataloging-in-Publication Data Adams, William W., 1937- An 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 1994 512/.4—dc20 94-19081 CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the 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. @ The 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 1.1. Introduction 1 1.2. The Linear Case 7 1.3. The One Variable Case 10 1.4. Term Orders 18 1.5. Division Algorithm 25 1.6. Grobner Bases 32 1.7. S-Polynomials and Buchberger's Algorithm 39 1.8. Reduced Grobner Bases 46 1.9. Summary 50

Chapter 2. Applications of Grobner Bases 53 2.1. Elementary Applications of Grobner Bases 53 2.2. Hilbert Nullstellensatz 61 2.3. Elimination 69 2.4. Polynomial Maps 79 2.5. Some Applications to 90 2.6. Minimal Polynomials of Elements in Field Extensions 97 2.7. The 3-Color Problem 102 2.8. Integer Programming 105

Chapter 3. Modules and Grobner Bases 113 3.1. Modules 113 3.2. Grobner Bases and Syzygies 118 3.3. Improvements on Buchberger's Algorithm 124 3.4. Computation of the Syzygy Module 134 3.5. Grobner Bases for Modules 140 3.6. Elementary Applications of Grobner Bases for Modules 152 3.7. Syzygies for Modules 161 3.8. Applications of Syzygies 171 3.9. Computation of Horn 183

Vll Vlll CONTENTS

3.10. Free Resolutions 194

Chapter 4. Grobner Bases over Rings 201 4.1. Basic Definitions 202 4.2. Computing Grobner Bases over Rings 212 4.3. Applications of Grobner Bases over Rings 225 4.4. A Primality Test 237 4.5. Grobner Bases over Principal Domains 246 4.6. Primary Decomposition in R[x] for R a PID 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 engi• neering 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 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). Com• puting is useful in producing and analyzing examples which illustrate a concept already understood, or which one hopes will give insight into a less well under• stood 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 pro-

IX 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 Buchberger1 [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 — gcd(/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,... ,xn] 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 elemen• tary 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 improve• ment 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 com• puting 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 com• puting 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 prepa• ration of the manuscript. Xll PREFACE

Chapter 2 Chapter 3 PREFACE xni

Chapter 4

3.2 4.1

4.2 | 2.4 | | 3.4 |

4.5 4.3

4.6 4.4 References

[AdGo] W.W. Adams and L.J. Goldstein, Introduction to Number Theory, Pren• tice Hall, Englewood Cliffs, NJ, 1976. [AtMD] M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Alge• bra, Addison-Wesley, Reading, MA, 1969. [Ba] D. Bayer, The Division Algorithm and the Hilbert Scheme, Ph.D. Thesis, Harvard University, Cambridge, MA, 1982. [BaSt88] D.A. Bayer and M. Stillman, On the complexity of computing syzygies, J. Symb. Comp. 6 (1988), 135-147. [BaSt90] D.A. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. User's manual. [BeWe] T. Becker and V. Weispfenning, Grobner Bases: A Computational Ap• proach to Commutative Algebra, Springer Verlag, Berlin and New York, 1993. [Bu65] B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. The• sis, Inst. University of Innsbruck, Innsbruck, Austria, 1965. [Bu79] B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Grobner bases, in EUROSAM'79, An International Sympo• sium on Symbolic and Algebraic Manipulation (E.W. Ng, ed.), Lecture Notes in Comput. Sci., vol. 72, Springer Verlag, Berlin and New York, 1979, 3-21. [Bu83] B. Buchberger, A note on the complexity of constructing Grobner bases, in EUROCAL'83, European Computer Algebra Conference (J.A. van Hulzen ed.), Lecture Notes in Comput. Sci., vol. 162, Springer Verlag, Berlin and New York, 1983, 137-145. [Bu85] B. Buchberger, Grobner bases: An algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory (N.K. Bose ed.), Reidel, Dordrecht, 1985, 184-232. [CGGLMW] B. Char, K. Geddes, G. Gonnet, B. Leong, M. Monogan, and S.M. Watt, Maple V Library Reference Manual, Springer Verlag, Berlin and New York, 1991. [Coh] H. Cohen, A Course in Computational , Springer Verlag, Berlin and New York, 1993. [CoTr] P. Conti and C. Traverso, Buchberger algorithm and integer programming,

279 280 REFERENCES

in Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes AAECC'9 (H.F. Mattson, T. Mora, and T.R.N Rao eds.), Lecture Notes in Comput. Sci., vol. 539, Springer verlag, Berlin and New York, 1991, 130- 139. [CLOS] D. Cox, J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer Verlag, Berlin and New York, 1992. [Cz] S.R. Czapor, Grobner basis methods for solving algebraic equations, Ph.D. Thesis, University of Waterloo, Canada, 1988. [EHV] D. Eisenbud, C. Huneke, and W. Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), 207-235. [FGLM] J.C. Faugere, P. Gianni, D. Lazard, and T. Mora, Efficient computation of zero-dimensional Grobner bases by change of ordering, J. Symb. Comp. 16 (1993), 329-344. [Ga] D. Gale, The Theory of Linear Economic Models, McGraw-Hill, New York, 1960. [GTZ] P. Gianni, B. Trager, and G. Zacharias, Grobner bases and primary de• composition of polynomial ideals, J. Symb. Comp. 6 (1988), 149-167. [GMNRT] A. Giovini, T. Mora, G. Niesi, L. Robbiano, and C. Traverso, uOne sugar cube, please" or selection strategies in the Buchberger algorithm, in Proc. International Symposium on Symbolic and Algebraic Computation ISSAC'91 (S.M. Watt ed.), ACM Press, New York, 1991, 49-54. [GiNi] A. Giovini and G. Niesi, Co Co A user manual, 1990. [Go] L. J. Goldstein, Abstract Algebra, a First Course, Prentice Hall, Englewood Cliffs, NJ, 1973. [He] I.N. Herstein, Topics in Algebra, Blaisdell, New York, 1964. [Hi] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math. 79 (1964), 109-326. [Him] T.W. Hungerford, Algebra, Springer Verlag, Berlin and New York, 1974. [Huy] D.T. Huynh, A superexponential lower bound for Grobner bases and Church-Rosser commutative Thue systems, Inf. Contr. 68 (1986), 196-206. [JeSu] R.D. Jenks and R.S. Sutor, AXIOM: The Scientific Computation System, Springer Verlag, Berlin and New York, 1992. [Lak] Y.N. Lakshman, On the complexity of computing Grobner bases for zero- dimensional polynomials ideals, Ph.D. Thesis, Rensselaer Polytechnic Insti• tute, Troy, NY, 1990. [Lan] S. Lang, Algebra, Addison-Wesley, Reading, MA, 1970. [Laz85] D. Lazard, Ideal bases and primary decomposition: Case of two vari• ables, J. Symb. Comp. 1 (1985), 261-270. [Laz91] D. Lazard, Systems of algebraic equations (algorithms and complexity), Computational Algebraic Geometry and Commutative Algebra (D. Eisenbud and L. Robbiano, eds.), Cambridge University Press, London, 1993. REFERENCES 281

[Mac] F.S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531-555. [MaMe] E.W. Mayr and A.R Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals. Adv. Math 46 (1982), 305- 329. [Mi] B. Mishra, Algorithmic Algebra, Springer Verlag, Berlin and New York, 1993. [M688] H.M. Moller, On the construction of Grobner bases using syzygies, J. Symb. Comp. 6 (1988), 345-359. [M690] H.M. Moller, Computing syzygies a la Gauss Jordan, in Effective Meth• ods in Algebraic Geometry (T. Mora and C. Traverso, eds.), Progress in Mathematics 94, Birkhauser Verlag, Basel, 1990, 335-345. [MoMo] H.M. Moller and T. Mora, New constructive methods in classical ideal theory, J. Algebra 100 (1986), 138-178. [MoRo] T. Mora and L. Robbiano, The Grobner fan of an ideal, J. Symb. Comp. 6 (1988), 183-208. [NZM] I. Niven, H.S. Zuckerman, and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th ed., Wiley, New York, 1991. [RoSw] L. Robbiano and M. Sweedler, Subalgebra bases, in Proc. Commutative Algebra Salvador (W. Burns and A. Simis eds.), Lecture Notes in Math., vol. 1430, Springer Verlag, Berlin and New York, 1988, 61-87. [Schre] F.O. Schreyer, Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrasschen Divisionsatz, Diplomarbeit, Hamburg, 1980. [Schri] A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chich• ester, NY, 1986. [Se] A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc. 197 (1974), 272-313. [ShSw] D. Shannon and M. Sweedler, Using Grobner bases to determine al• gebra membership, split surjective algebra homomorphisms determine bira- tional equivalence, J. Symb. Comp. 6 (1988), 267-273. [Sz] G. Szekeres, A canonical basis for the ideals of a polynomial domain, Am. Math. Mon. 59 (1952), 379-386. [Wo] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, Reading, MA, 1991. [Za] G. Zacharias, Generalized Grobner bases in commutative polynomial rings, Bachelor's Thesis, MIT, Cambridge, MA, 1978. List of Symbols

N natural numbers Z ring of integers

Zn ring of integers modulo n Q field of rational numbers M field of real numbers C field of complex numbers n T set of power products in the variables x\,... , xn k a field kn affine space k[x\,... ,xn] ring of polynomials in the variables sci,... ,xn with coefficients in the field k

{fii • • • > fs) ideal (submodule) generated by /i,... , /s / = g (mod J) / congruent to g modulo / k[x\,... , xn]/I quotient ring of k[x\,... , xn] by the ideal J / -f- / coset of / modulo J V(S) variety in kn defined by the set of polynomials S n V{fi,... , fa) variety in k defined by the polynomials /i,... , fs n I(V) ideal in k[x\,... , xn] defined by V C k (S) ideal generated by the polynomials in the set S gcd greatest common divisor lcm least common multiple dim*; (L) dimension of the A;-vector space L lt(/) leading term of / lp(/) leading power product of / lc(/) leading coefficient of / ai a X <* h f reduces to h modulo G in one step f —>+ h f reduces to h modulo G

283 284 LIST OF SYMBOLS

f^h f reduces to h and h = f — Xg U(S) leading term ideal (submodule) defined by S S(f,g) S-polynomial of / and g NGU) normal form of / with respect to G n VK(S) variety in K denned by S C k[xi,... ,xn] k algebraic closure of the field k radical of the ideal J inj intersection of J and J I: J ideal quotient of J by J im(4>) image of the map 0 ker() kernel of the map (j)

k[h,... fs] A:-algebra generated by the polynomials /1,... , fs k[F] fc-algebra generated by the polynomials in the set F

K\Xi, . . . , Xn) rational function field k(a) field extension of k obtained by adjoining a

k(ai,... ,an) field extension of k obtained by adjoining a\,... ,OLn set of column vectors with entries in the ring A M/N quotient module of M by N isomorphic to m 61,... , en standard basis for the free module A [fi '•• fs ] 1x5 matrix whose entries are polynomials /1,.. • 1 Js [A - fs] m x 5 matrix whose columns are vectors /l5... ,fs svzv Syz(/i,... fs) gy module of the matrix [ f\ • • • fs ] Syz(F) syzygy module of the matrix F M/) leading monomial of the vector / Fee direct sum of the matrices F and G [F\G] concatenation of the matrices F and G *S transpose of the matrix S ann(M) annihilator of the module M F®G tensor product of the matrices F and G Hom(M, AT) set of all A-module homomorphisms between M and N (U) module generated by the columns of the matrix U S-XA localization of the ring A at the multiplicative set S localization of the ring A at the prime ideal P AP localization of the ring A at the set {1, g, g2, #3,... } Aa Index

affine algorithm for modules, 149 algebra, 84, 180 improved algorithm, 129 space, 2 Theorem, 40, 48 algebra, 232 affine, 84, 180 CoCoA, 275 First Isomorphism Theorem, 80 column vector, 113 homomorphism, 79, 92, 232 commutative ring, 1 membership problem, 39 computation, 275 algebraic Computer Algebra System, vii, 275 closure, 62 confluence relation, 37 element, 97 congruence, 5 geometry, 90 Conti, 105 algebraically closed field, 62 coset, 5 algorithm, 275 field case crit2, 130 multivariable case, 57 division one variable case, 15 module case, 145 module case, 155 multivariable case, 28 ring case, 227 one variable case, 12 cost function, 105 ring case, 207 critl, 128, 152, 258 Euclidean, 14 crit2, 128, 168, 224 Grobner basis Buchberger's, field case, 43 Buchberger's, module case, 149 deglex, 20 improved Buchberger's, 129 degrevlex, 20 PID case, 250 determine, 5, 53 ring case, 216 Dickson's Lemma, 23 ring case using Moller's techniqi direct sum of matrices, 174 219 division primality test, 244 for vectors, 141 annihilator, 177, 217 multivariable case, 25, 26, 27 ascending chain condition, 6 one variable case, 11 division algorithm basis field case for Am/M, 155 multivariable case, 28 one variable case, 12 for k[xi,... ,xn]//, 58 in Am, 114 module case, 145 standard in Am, 114 ring case, 207 Bayer, 102 Buchberger, viii, 39, 113, 124, 131 effective, 53 algorithm, ix, 43 effective coset representatives, 226

285 286 INDEX elimination for ideal, ring case, 201, 208 Gauss-Jordan, 2 minimal, 211 ideal minimal strong, 251, 254, 262 field case, 70 strong, 251 ring case, 229 for module, 147 module, 156 reduced, 150 order, 69, 229 Euclidean Algorithm, 14 Hensel's Lemma, 267 exact sequence, 185 Hilbert short, 184 Basis Theorem, 5, 6 explicitly given module, 117 Nullstellensatz, 62 Ext functor, 194, 199 Horn, 183 homogeneous Faugere, 68 ideal, 49 field component, 22 algebraic closure, 62 polynomial, 22, 38 algebraically closed, 62 syzygy, 121, 212 First Isomorphism Theorem homogenization, 38 algebra case, 80 homological algebra, 185 module case, 117 homomorphism free module, 114 algebra, 79, 92, 232 free resolution, 195 image of, 80 kernel of, 80, 232 Gauss-Jordan elimination, 2 module, 116 generating set, 3, 114 finite, 6 ideal, 3 Gianni, 68, 237 field case global dimension, 196 elimination, 70 graph, 102 Grobner basis for, 32, 34 graph of a map, 91 homogeneous, 49 greatest common divisor, 13, 15, 71 intersection, 70, 172, 175 Grobner basis leading term, 32 algorithm membership problem, 5, 53 Buchberger's, field case, 43 monomial, 23 Buchberger's, module case, 148 of relations, 80 improved Buchberger's, 129 quotient, 72 PID case, 250 quotient for modules, 157, 177, 182 ring case, 216 irreducible, 261 ring case using Moller's technique, leading term, 32, 206 219 monomial, 23 for ideal, field case, 32, 34 P- primary, 261 minimal, 47 primary, 260 reduced, 48 prime, 61, 237, 259 universal, 50 principal, 13, 246 using syzygies, 121 radical of, 62, 259 INDEX 287

ring case linear equations are solvable, 204, 262 elimination, 229 local ring, 238 Grobner basis for, 201, 208 localization, 89 intersection, 230 long division, 10 leading term, 206 membership problem, 225 membership problem of relations, 232 algebra, 39 quotient, 217, 231 ideal over field, 5, 53 saturation of, 238 ideal over ring, 225 zero-dimensional, 64, 262 module, 153 image of a homomorphism, 80 minimal implicitization, 91 Grobner basis, field case, 47 induction, 277 Grobner basis, ring case, 211 integer programming, 105 strong Grobner basis, 251 inter-reduced, 131 with respect to F, 205 intersection minimal polynomial, 97 of ideals module, 113 field case, 70, 172, 175 elimination, 156 ring case, 230 explicitly given, 117 of modules, 157, 175, 176, 178 First Isomorphism Theorem, 117 inverse in k[xi,... ,xn]/7, 59, 182 free, 114 irreducible ideal, 261 Grobner basis for, 147 isomorphic varieties, 94 homomorphism, 116 isomorphism, module, 116 ideal quotient, 157, 177, 182 intersection, 157, 175, 176, 178 kernel of a homomorphism, 80, 232 isomorphism, 116 leading term module, 147 Lakshman, 77 membership problem, 153 Lazard, 68, 201, 254, 259 Noetherian, 116 leading coefficient normal form, 155 multi-variable case, 21, 202 quotient, 116 one variable case, 10 reduction in, 143 vector case, 143 S-polynomial, 148 leading monomial, 143 syzygy, 118 leading power product, 21, 202 Moller, 170, 201, 216, 250 leading term monic polynomial, 13 ideal, 32, 206 monoid, 37 module, 147 monomial multivariate case, 21, 202 ideal, 23 one variable case, 10 in modules, 141 vector case, 143 leading, 143 least common multiple Mora, 23, 68 of polynomials, 71 multiplicative set, 238 of vectors, 147 lex, 19 Noetherian 288 INDEX

module, 116 ideal, 61, 237, 259 ring, 6, 259 principal ideal, 13, 246 normal form Principal Ideal Domain (PID), 13, 246 polynomial case projection map, 90 over a field, 57 pullback, 182 over a ring, 227 vector case, 155 quotient normal selection strategy, 130 module, 116 Nullstellensatz, 62 ring, 5 order, 18 radical of an ideal, 62, 259 degree lexicographic (deglex), 19 reduced degree reverse lexicographic Grobner basis (degrevlex), 20 ideal case over fields, 48 elimination, 69 module case, 150 induced by a matrix, 166 polynomial over a field, 27 lexicographic (lex), 19 totally, 227 position over term (POT), 142 vector, 144 term, 18, 140 reduction term over position (TOP), 142 linear case, 8 total, 18, 277 module case, 143 well-ordering, 18, 277 multivariable case over a field, 25, 26, 27 P-primary ideal, 261 multivariable case over a ring, 203, 205 parametrized variety, 91 one variable case, 11 polynomial, 1 strong, 252, 258 elementary symmetric, 25 remainder homogeneous, 22, 38 module case, 144 minimal, 97 multivariable case over a field, 27 monic, 13 one variable case over a field, 11 normal form, 57, 227 ring reduced, 27 commutative, 1 square free, 75 local, 238 symmetric, 25, 88 localization, 89 position over term (POT), 142 Noetherian, 6, 259 power product, 1, 18 of fractions, 238 leading, 21, 202 quotient, 5 presentation, 117, 119, 195 Robbiano, 23, 39 primality test, 244 row echelon form, 2 primary component, 261 S-basis, 247 decomposition, 261 S-polynomial ideal, 260 module case, 148 prime multivariable case over a field, 40, 121, component, 261 124 INDEX 289

PID case, 249 universal Grobner basis, 50 ring case, 213 SAGBI basis, 39 variety, 2, 61, 90 saturated set, 213, 226 isomorphic, 94 saturation of a set, 213 parametrized, 91 saturation of an ideal, 238 vector space, 1 Schreyer, 165 Seidenberg, 78 well-ordering, 18, 277 Shannon, 82, 88 square free polynomial, 75 Zacharias, 201, 226, 237 standard basis, 32 Zariski closure, 90 strong zero divisor, 61 Grobner basis, 251 zero-dimensional ideal, 64, 262 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 (TOP), 142 three color problem, 102 total order, 18, 277 totally reduced, 227 Trager, 237 transpose matrix, 176 Traverso, 105