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53 David A. Cox and Bernd Sturmfels , Editors, Applications of computational algebraic geometry (San Diego, California, January 1997)
52 V . Mandrekar and P . R . Masani , Editors, Proceedings of the Norbert Wiener
Centenary Congress, 1994 (East Lansing, Michigan, 1994)
51 Louis H. Kauffman, Editor, The interface of knots and physics (San Francisco,
California, January 1995)
50 Robert Calderbank, Editor, Different aspects of coding theory (San Francisco,
California, January 1995)
49 Robert L. Devaney , Editor, Complex dynamical systems: The mathematics behind the
Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994)
48 Walter Gautschi , Editor, Mathematics of Computation 1943-1993: A half century of
computational mathematics (Vancouver, British Columbia, August 1993)
47 Ingrid Daubechies , Editor, Different perspectives on wavelets (San Antonio, Texas,
January 1993)
46 Stefan A. Burr, Editor, The unreasonable effectiveness of number theory (Orono,
Maine, August 1991)
45 D e W i t t L. Sumners , Editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992)
44 Be la Bol lobas, Editor, Probabilistic combinatorics and its applications (San Francisco,
California, January 1991)
43 Richard K. Guy , Editor, Combinatorial games (Columbus, Ohio, August 1990)
42 C. Pomerance , Editor, Cryptology and computational number theory (Boulder,
Colorado, August 1989)
41 R. W . Brockett , Editor, Robotics (Louisville, Kentucky, January 1990)
40 Charles R. Johnson, Editor, Matrix theory and applications (Phoenix, Arizona, January 1989)
39 Robert L. Devaney and Linda Keen , Editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988)
38 Juris Hartmanis , Editor, Computational complexity theory (Atlanta, Georgia, January
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37 Henry J. Landau, Editor, Moments in mathematics (San Antonio, Texas, January 1987)
36 Carl de Boor , Editor, Approximation theory (New Orleans, Louisiana, January 1986)
35 Harry H. Panjer, Editor, Actuarial mathematics (Laramie, Wyoming, August 1985) 34 Michael Anshel and Wil l iam Gewirtz , Editors , Mathematics of information
processing (Louisville, Kentucky, January 1984) 33 H. P e y t o n Young, Editor, Fair allocation (Anaheim, California, January 1985) 32 R. W . McKelvey , Editor, Environmental and natural resource mathematics (Eugene,
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31 B . Gopinath , Editor, Computer communications (Denver, Colorado, January 1983)
30 S imon A. Levin, Editor, Population biology (Albany, New York, August 1983)
29 R. A . DeMil lo , G. I. Davida, D . P. Dobkin , M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981)
28 R. Gnanadesikan, Editor, Statistical data analysis (Toronto, Ontario, August 1982)
27 L. A. Shepp, Editor, Computed tomography (Cincinnati, Ohio, January 1982)
26 S. A . Burr, Editor, The mathematics of networks (Pittsburgh, Pennsylvania, August
1981)
25 S. I. Gass , Editor, Operations research: mathematics and models (Duluth, Minnesota,
August 1979)
24 W . F. Lucas, Editor, Game theory and its applications (Biloxi, Mississippi, January 1979)
(Continued in the back of this publication)
http://dx.doi.org/10.1090/psapm/053
AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures
published as a subseries of Proceedings of Symposia in Applied Mathematics
Proceedings of Symposia in
APPLIED MATHEMATICS
Volume 53
Applications of Computational Algebraic Geometry
American Mathematical Society Short Course January 6-7, 1997 San Diego, California
David A. Cox and Bernd Sturmfels, Editors
Dinesh Manocha Thomas W. Sederberg Xenia H. Kramer and Reinhard C. Laubenbacher Rekha R. Thomas John B. Little
American Mathematical Society !f Providence, Rhode Island
L E C T U R E N O T E S P R E P A R E D F O R T H E
A M E R I C A N M A T H E M A T I C A L S O C I E T Y S H O R T C O U R S E
A P P L I C A T I O N S O F C O M P U T A T I O N A L A L G E B R A I C G E O M E T R Y
H E L D IN S A N D I E G O , C A L I F O R N I A
J A N U A R Y 6 - 7 , 1997
T h e AMS Short Course Series is sponsored by the Society's Program Commi t t ee for National Meetings. T h e series is under the direction of the Short Course
Subcommi t tee of t h e Program Commi t t ee for National Meetings.
1991 Mathematics Subject Classification. P r i m a r y 13F99, 14Q99; Secondary 05E99, 68Q40, 68U07, 90C10, 94B27.
Library of Congress Cataloging-in-Publicat ion D a t a
Applications of computational algebraic geometry : American Mathematical Society short course, January 6-7, 1997 / David A. Cox and Bernd Sturmfels, editors ; Dinesh Manocha . . . [et al., authors].
p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634 ; v. 53. AMS short course lecture notes)
Includes bibliographical references and index. ISBN 0-8218-0750-1 (alk. paper) 1. Geometry, Algebraic—Data processing—Congresses. I. Cox, David A. II. Sturmfels,
Bernd, 1962- . III. Manocha, Dinesh N. IV. Series: Proceedings of symposia in applied mathematics ; v. 53. V. Series: Proceedings of symposia in applied mathematics. AMS short course lecture notes. QA564.A59 1997 516.3'5—dc21 97-29117
CIP
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10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98
Contents
Preface ix
Introduction to Grobner bases DAVID A. Cox 1
Introduction to resultants BERND STURMFELS 25
Numerical methods for solving polynomial equations DlNESH MANOCHA 41
Applications to computer aided geometric design THOMAS W. SEDERBERG 69
Combinatorial homotopy of simplicial complexes and complex information systems XENIA H. KRAMER AND REINHARD C. LAUBENBACHER 93
Applications to integer programming REKHA R. THOMAS 119
Applications to coding theory JOHN B. LITTLE 143
Index 169
Preface
The last 30 years have witnessed a dramatic expansion in our ability to manipulate polynomial equations. Beginning with the discovery of Grobner bases in the 1960s and fueled by the advent of modern computers and a renewed appreciation of resultants, computational algebraic geometry has grown rapidly in importance. It has had a profound impact on algebraic geometry, where new theoretical methods and an enhanced ability to compute examples have led to significant new results. At the same time, computational algebraic geometry possesses a down-to-earth quality which makes it unusually applicable and accessible. In recent years, Grobner bases, resultants and other computational tools have been applied to a wide variety of problems dealing with systems of polynomial equations. In general, the ability to "crunch equations" as easily as we "crunch numbers" has led to a rethinking of how computers are used in research, industry and education.
The lectures in this volume were first presented at the Short Course on Applications of Computational Algebraic Geometry given in San Diego, CA on January 6-7, 1997. They were written to bring the basic ideas of computational algebraic geometry to a wide audience of mathematicians. The first two lectures introduce the two key players in the subject, Grobner bases and resultants, and the third lecture surveys some recent methods for solving polynomial equations. The final four lectures discuss applications to computer aided geometric design, complex information systems, integer programming, and coding theory. Our hope is that these lectures will convince you of the beauty and utility of this subject and encourage you to learn more about it.
We would like to thank the AMS for their support and wish to express our gratitude to the AMS staff who helped so ably during the Short Course and later during the preparation of this volume.
David A. Cox Bernd Sturmfels
IX
Index
3-coloring, 12, 13
A-contractible, 98, 108, 117 A-homotopy, 92, 95-98, 106-109, 111, 113,
115-117 A-homotopy group, 98 abelian code, 152 affine space, 32 affine variety, 5, 122 algebraic coding theory, see coding theory algebraic curve, 43, 54, 56, 162, 163 algebraic numbers, 21, 30 automorphism, 164-166 Axiom, 9, 19
Bezier curve, 67-69, 73-75, 77, 78, 82-84, 86, 87
Bezout bound, 45, 47, 51 Bezout resultant, 26, 28, 29, 71-73 Bezout's Theorem, 32, 45, 70, 83, 88 backsolving, 2, 11 base points, 79, 83, 89 base simplex, 98, 106, 109 basis conversion, 13, 159 Berlekamp-Massey algorithm, 153, 155, 159-
161, 166 Bernstein basis, 43 Bernstein bound, 45, 47, 51 Bernstein polynomial, 68, 73, 84, 85 Bernstein's Theorem, 36 bicubic patch, 87, 88 binomial, 111, 112, 114, 115, 122, 123, 125,
126, 130-132 block code, 143, 144, 149 boat conformation, 61, 62 bond angles, 59-62 bond lengths, 59, 60-62 boundary representation, 62, 63 Buchberger's algorithm, 8, 27, 98, 102, 103,
121, 123, 124, 126, 140, 155, 159 Buchberger's criterion, 8
CAGD, 41-43, 67, 70, 78, 80, 82, 87, 89 Canny-Emiris matrix, 37, 38 Cayley-Koszul complex, 38 CD digital audio system, 143, 146, 149 chair conformation, 60-62
characteristic polynomial, 44, 45 Chow form, 38 circuit, 129-132 closed component, 52, 53 Cocoa, 26, 42 codeword, 144-147, 149, 150, 152-155, 160,
164 coding theory, 9, 143, 159 Color Plate, 62, 63, 87, 88 combinatorial homotopy, 91, 92, 110, 116,
see also A-homotopy companion matrix, 16 complete graph, 131 complex information systems, 9, 91, 117 complex projective space, 32 complexity, 3, 13, 21, 22, 38, 43, 45, 63, 121,
166 component jumping, 54 component splitting, 54, 56 comprehensive Grobner basis, see Grobner
basis computer aided geometric design, see CAGD computer algebra system, 2, 5, 9, 19, 41, 42 computer graphics, 41 computer vision, 41, 43 conformation, 59 conformers, 57 constructive solid geometry, 62 Conti-Traverso algorithm, 123-125, 139 control points, 68, 73, 84 control polygon, 68, 69, 85 convex hull, 84, 122 coset representatives, 7, 14 cost function, 119, 120, 128, 136 cost vector, 119, 120, 122, 123, 130, 131, 134,
135, 138, 140 CPLEX, 140 Cramer's rule, 131 cubic surface, 80, 87, 88 curve components, 52 curve intersection, 62, 63, 83 curve parametrization, 75, 77 cutting plane, 120, 121 cyclic codes, 148, 149, 151, 152, 161, 164,
166 cyclohexane, 60-62
170 INDEX
cylindrical algebraic decomposition, 43
de Casteljau algorithm, 86 decoding, 144, 149, 153, 154, 156, 157, 159,
160, 166 deg-lex, 99, 100, 104, 105 degree elevation, 69 degree of the resultant, 32 degree-lexicographic, see deg-lex Descarte's rule of signs, 85 DH formulation, 59 DH parameters, 59, 60 difference operations, 62 dihedral angles, 59-62 division algorithm, 3-6, 152, 153 divisor, 162, 163 double point, 75, 77, 83
eigenvalues, 3, 16, 17, 19, 20, 43-46, 49-51, 54-56, 58, 60, 84
Eigenvalue Theorem, 16, 17 eigenvectors, 16, 18-20, 45, 49, 56 EISPACK, 45, 46 elbow, 88 eliminate, 51 eliminating, 47 elimination ideal, 10, 11, 80 elimination order, 12, 124 Elimination Theorem, 10, 11, 15, 25 elimination theory, 3, 9, 25-27, 31, 38, 44 encoding, 144, 146, 147, 149, 166 error locations, 154, 157, 161 error locator ideal, 161 error locator polynomial, 155, 157, 161 error polynomial, 154, 156 error values, 154, 157 error vector, 144 essential supports, 35, 36 extended Reed-Solomon codes, 148 Extension Theorem, 11, 13 extraneous factor, 34
/-vector, 92-95 faces, 92 feasible, 119, 120, 124, 125, 138 Fermat's Last Theorem, 78 finite d-representation, 103, 104 finite field, 143, 148, 162 Finiteness Theorem, 13, 20 First Eigenvector Theorem, 18, 20 floating point arithmetic, 42, 83-87 free associative algebra, 92, 98 free resolutions, 22 fundamental group, 107, 108, 110, 117
Gk continuity, 88 Gaussian elimination, 46, 55 generalized companion matrix, 17 generalized eigenvalue, 46, 49, 56
generator matrix, 146, 147, 149 generator polynomial, 150, 151, 166 generic cost vectors, 134, 135, 138 generic polynomials, 45 genus, 75, 76, 78, 83, 162, 163 geometric modeling, see CAGD Gilbert-Varshamov bound, 146, 161 Goppa codes, 162, 163, 165, 166 graded reverse lexicographic, see grevlex Graver arrangement, 139 Graver basis, 127-132, 138 greatest common divisor, 34 grevlex, 3, 9, 12, 21 GRIN, 124, 140 Grobner arrangement, 138, 139 Grobner basis, 1, 3, 5-14, 17, 19, 21, 22, 2 5 -
27, 30, 42, 43, 70, 71, 73, 80, 92, 98, 100-106, 108, 109, 111-113, 115, 116, 121-125, 127, 128, 131, 132, 137-140, 152-154, 158-161, 165, 166
comprehensive, 22, 154 reduced, see also reduced Grobner basis universal, 22, 128, 130, 131, 137-139
Grobner basis of a module, 22, 154, 157, 165 Grobner cone, 135, 137, 139 Grobner fan, 134-136, 138, 139 Grobner fiber, 134-136
Hamming distance, 144, 147 Hasse-Weil bound, 163, 164 Hermitian curve, 164-166 Hessenberg form, 46 Hilbert basis, 128 Hilbert Basis Theorem, 6, 123 Hilbert Nullstellensatz, 12, 15 history of Grobner bases, 9 homogeneous polynomials, 31 homotopy methods, 42, 53 HOMPACK, 42 hyperplane arrangement, 133, 136
ideal of leading terms, 6 ideal, 5 implicit equation, 70, 71, 73-75, 79-83, 86,
87 implicit representation, 11 implicit surface, 88 implicitization, 11, 22, 29, 67, 70, 73, 80 improperly parametrized curves, 72, 73 incidence matrix, 93 incidence variety, 32 index of speciality, 162 information flow, 91, 117 information positions, 147, 153, 166 information rate, 145 initial ideal, 125 initial term, 4 inner normal cone, 132 inner normal fan, 133
INDEX 171
integer programming, 9, 120-122, 124, 125, 127, 130, 131, 134, 138-140
interpolation basis, 15, 18 intersection curve, see curve intersection interval analysis, 43 inverse power iterations, 52, 55, 56 inversion, 67, 71, 72, 74, 87 isolines, 54
Jenkins-Traub algorithm, 41 joint angles, 58
key equation, 156-159, 161 kinematics, 44, 45, 56-58, 60
Lagrange multipliers, 1 LAPACK, 41, 45, 46 Laurent polynomials, 35 Lawrence lifting, 128, 136 leading coefficient, 99 leading monomial, 7, 99, 105, 115, 125 leading term, 4, 6, 99, 123, 125, 139, 158 length-lexicographic, 99 lex, 3, 9, 10, 12, 21, 154 lexicographic, see lex linear algebra, 44, 45, 51, 56 linear block codes, 146 linear code, 147, 162 linear programming, 119-121, 130, 131, 138 linear relaxation, 120, 140 linear system, 79 links, 58 LINPACK, 41 loop, 52, 53 LQ factorization, 55 LU decomposition, 55
m-dimensional cyclic code, 152 Macaulay, 9, 26, 42, 124, 139 MACSYMA, 9 Magma, 9 Main Theorem of Elimination Theory, 32 Maple, 2, 5, 9, 19, 26, 29, 42 Mathematica, 2, 5, 9, 19, 42 matrix polynomial, 47-50, 56 MDS codes, 148 minimal element, 158-161 minimal polynomial, 17, 30 Minkowski integral, 133, 135, 136 Minkowski sum, 133, 134, 136, 137 Minkowski summand, 133, 134 mixed subdivision, 37 mixed volume, 36 molecular chain, 56, 57, 59, 60 molecular modeling, 41, 57 molecules, 44 monomial ideals, 128, 134 monomial order, 3, 6, 152, 157-159, 165, see
also term order
Mora algorithm, 103, 104, 111, 112, 115 moving surface, 80, 82 multiplicity, 16, 20, 51, 75 multipolynomial or multivariable resultant,
21, 31, 34, 37, 44, 48, 51
nearest-neighbor, 144, 145 Newton polygons, 36 Newton poly types, 37 Newton's method, 42, 53 nomogram, 74, 75 normal cone, see inner normal cone normal fan, see inner normal fan normal form, 101, 109, 110, 123, 124 normal monomials, 101, 116 NP-complete, 13 number of solutions, 14, 15
offset curves, 77, 78 one-dimensional solutions, 41, 43, 44, 51, 56,
62 open components, 52, 56 optimal cost value, 119, 122 optimal solution, 119-127, 130, 131, 134, 140 optimization problems, 119 oracle, 127 overlaps, 102, 103
Pade approximation, 156 parametric equation, 67, 70, 71, 77, 78, 86,
88 parametric representation, 11 parametric surface, 62, 80, 81 parity check, 147, 152, 153, 166 partial solution, 11, 13 perfect /-matching problem, 131 polyhedral cone, 128, 132 polyhedral fan, 132 polyhedron, 119, 132, see also polytope polynomial consequence, 5 polytope, 22, 119-122, 126, 127, 132, 133,
135-137 power method, 46, 47 primitive element, 146 primitive vector, 129 protein folding, 57 pseudohomotopy, 92, 95 Pythagorean triples, 78 Pythagorean-hodograph curves, 77, 78
Q-analysis, 92 Q-matrix, 94, 95 Q-vector, 92, 94, 95 q-chain, 95-98, 106, 107, 109, 113, 114, 117 ^-connected, 93, 107, 109, 117 g-loops, 98, 106, 107 g-near, 93, 97, 109 QR algorithm, 50 QZ algorithm, 46, 50
172 INDEX
radical, 15 radical ideal, 19 Radical Theorem, 15, 17, 18 rational curve, 75 Rational Implicitization Theorem, 71, 80, 82 rational parametric surfaces, 62 ray tracing, 87 Reduce, 9, 42 reduced Grobner basis, 9, 12, 20, 106, 115,
123-126, 128, 129, 134, 136-139, 158 reducible curve, 75 Reed-Solomon code, 146-151, 153, 154, 160,
162, 163, 166 refinement, 133 remainder, 4, 7, 8, 17, 150, 152 resultant, 21, 22, 27, 32, 33, 34, 38, 43-45,
47, 49, 63, 67, 70, 73, 80, 83, 84 revolute joints, 56, 58 Riemann-Roch theorem, 162, 163 ring closure, 59, 60 robotics, 41, 43, 44, 56-58
S-polynomial, 7, 103-105, 111, 112, 123 Second Eigenvector Theorem, 19, 20 set of matches, 102 Shannon's Theorem, 145 simplex algorithm, 119, 120, 138 simplices, 92 simplicial complex, 91-93, 95, 98, 106, 116,
117 simplicial family, 92-94, 106, 107 simultaneous eigenvectors, 17, 18 Singleton bound, 145-148, 163 SINGULAR, 9, 26 singular points, 52, 54 singular set, 52 singular value decomposition, 46 skeleton, 95 solid modeling, 43, 44, 56, 62, 63, see also
CAGD solving systems of polynomial equations, 3,
9, 11, 13, 16, 17, 20-22, 27, 41, 43, 63 spanning tree, 108-111 sparse matrices, 63 sparse elimination theory, 35, 45 sparse matrix computations, 47 sparse resultant, 21, 35-38, 58 standard expression, 4 standard monomials, 125, 153, 166 state polytope, 134-138 Steiner surface, 81, 83 stochastic constraint, 127 stochastic integer programs, 127 Sturm sequences, 41 Sturm's Theorem, 21 submarine, 63 subresultants, 38 support of a polynomial, 35 support of a vector, 122, 123, 129
surface intersection, 82, 83, 87 surface of revolution, 82 Sylvester resultant, 25, 26, 28-30, 42, 44, 71 Sylvester-type formula, 37 syndrome polynomial, 155, 160 syndromes, 154, 155 systematic encoding, 147, 152, 153 syzygies, 22
tangent continuity, 88 teapot, 87 term, 4 term order, 4, 38, 98-101, 106, 109, 115,
122-124, see also monomial order test set, 125-127, 130, 140 toric ideal, 122, 126, 128, 131, 139, 140 toric variety, 122 tracing algorithm, 52-55 twisted boat conformation, 60-62
unimodular matrix, 131, 132 universal Grobner basis, see Grobner basis universal test set, 128, 130 Uspensky's method, 41
vertices, 92
Wilkinson polynomial, 85
zero-dimensional solutions, 41, 42, 44, 47, 56, 61
zonotope, 133, 136
Selected Titles in This Series (Continued from the front of this publication)
23 R. V. Hogg, Editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980)
22 G. H. Golub and J. Oliger, Editors, Numerical analysis (Atlanta, Georgia, January
1978)
21 P. D . Lax, Editor, Mathematical aspects of production and distribution of energy (San
Antonio, Texas, January 1976)
20 J. P. LaSalle, Editor, The influence of computing on mathematical research and
education (University of Montana, August 1973)
19 J. T. Schwartz, Editor, Mathematical aspects of computer science (New York City,
April 1966)
18 H. Grad, Editor, Magneto-fluid and plasma dynamics (New York City, April 1965)
17 R. Finn, Editor, Applications of nonlinear partial differential equations in mathematical
physics (New York City, April 1964) 16 R. Bel lman, Editor, Stochastic processes in mathematical physics and engineering (New
York City, April 1963) 15 N . C. Metropol is , A. H. Taub, J. Todd, and C. B. Tompkins, Editors,
Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962)
14 R. Bel lman, Editor, Mathematical problems in the biological sciences (New York City, April 1961)
13 R. Bel lman, G. Birkhoff, and C. C. Lin, Editors, Hydrodynamic instability (New York City, April 1960)
12 R. Jakobson, Editor, Structure of language and its mathematical aspects (New York City, April 1960)
11 G. Birkhoff and E. P. Wigner , Editors, Nuclear reactor theory (New York City, April 1959)
10 R. Be l lman and M. Hall, Jr., Editors, Combinatorial analysis (New York University, April 1957)
9 G. Birkhoff and R. E. Langer, Editors, Orbit theory (Columbia University, April
1958)
8 L. M. Graves, Editor, Calculus of variations and its applications (University of Chicago,
April 1956)
7 L. A. MacColl , Editor, Applied probability (Polytechnic Institute of Brooklyn, April
1955)
6 J. H. Curtiss , Editor, Numerical analysis (Santa Monica City College, August 1953)
5 A. E. Heins , Editor, Wave motion and vibration theory (Carnegie Institute of
Technology, June 1952)
4 M. H. Martin, Editor, Fluid dynamics (University of Maryland, June 1951)
3 R. V. Churchill, Editor, Elasticity (University of Michigan, June 1949)
2 A. H. Taub, Editor, Electromagnetic theory (Massachusetts Institute of Technology, July 1948)
1 E. Reissner, Editor, Non-linear problems in mechanics of continua (Brown University, August 1947)
ISBN 0-8218-0750-1
