Solving System s of Polynomial Equation s

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Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Series in Mathematics

Number 9 7

Solving System s of Polynomial Equation s

Bernd Sturmfel s

Published fo r th e Conference Boar d o f th e Mathematica l Science s

by th e t£2=S& America n Mathematica l Soc ie t y 0 ^ L «

wc Providence , Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n "^NJPA^

http://dx.doi.org/10.1090/cbms/097

CBMS Conferenc e o n Solvin g Polynomia l Equat ion s hel d a t Texas A& M University , Colleg e Stat ion , Texa s

May 20-24 , 200 2

Par t ia l ly suppor te d b y th e Nat iona l Scienc e Foundat io n

2000 Mathematics Subject Classification. P r imar y 13P10 , 14Q99 , 65H10 ; Secondary 12D10 , 14P10 , 35E20 , 52B20 , 62J12 , 68W30 , 90C22 , 91A06 .

Maple® - Waterlo o Maple , Inc. , Ontario , Canad a

MATLAB® - Th e MathWorks , Inc. , Natick , M A

Library o f Congres s Cataloging-in-Publicatio n D a t a

CBMS Conferenc e o n Solvin g Polynomia l Equation s (200 2 : Texa s A& M University ) Solving system s o f polynomia l equation s / Bern d Sturmfels .

p. cm . — (Conferenc e Boar d o f th e Mathematica l Science s regiona l conferenc e serie s i n mathematics, ISS N 0160-764 2 ; no. 97 )

Includes bibliographica l reference s an d index . ISBN 0-8218-3251- 4 (alk . paper ) 1. Equations-Numerical solutions-Congresses . 2 . Polynomials-Congresses. I . Sturmfels, Bernd ,

1962- II . Titl e III . Regiona l conferenc e serie s i n mathematic s ; no. 97 .

QA1.R33 no . 9 7 [QA214] 510 s-dc2 1 200202795 1 [512.9/42]

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Contents

Preface

Chapter 1 . Polynomial s i n On e Variabl e 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

The Fundamenta l Theore m o f Algebr a Numerical Roo t Findin g Real Root s Puiseux Serie s Hypergeometric Serie s Exercises

Chapter 2 . Grobne r Base s o f Zero-Dimensiona l Ideal s 2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

Computing Standar d Monomial s an d th e Radica l Localizing an d Removin g Know n Zero s Companion Matrice s The Trac e For m Solving Polynomia l Equation s i n Singula r Exercises

Chapter 3 . Bernstein' s Theore m an d Fewnomial s 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

From Bezout' s Theore m t o Bernstein' s Theore m Zero-dimensional Binomia l System s Introducing a Tori c Deformatio n Mixed Subdivision s o f Newton Polytope s Khovanskii's Theore m o n Fewnomial s Exercises

Chapter 4 . Resultant s 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

The Univariat e Resultan t The Classica l Multivariat e Resultan t The Spars e Resultan t The Unmixe d Spars e Resultan t The Resultan t o f Four Trilinea r Equation s Exercises

Chapter 5 . Primar y Decompositio n 5.1. 5.2. 5.3. 5.4. 5.5.

Prime Ideals , Radica l Ideal s an d Primar y Ideal s How to Decompos e a Polynomia l Syste m Adjacent Minor s Permanental Ideal s Exercises

vii

1 1 3 5 6 8

11

13 13 15 17 20 23 26

29 29 32 33 35 38 41

43 43 46 49 52 55 57

59 59 61 63 67 69

V

vi C O N T E N T S

Chapter 6 . Polynomia l System s i n Economic s 7 1 6.1. Three-Perso n Game s wit h Tw o Pur e Strategie s 7 1 6.2. Tw o Numerica l Example s Involvin g Squar e Root s 7 3 6.3. Equation s Definin g Nas h Equilibri a 7 7 6.4. Th e Mixe d Volum e o f a Produc t o f Simplice s 7 9 6.5. Computin g Nas h Equilibri a wit h PHCpac k 8 2 6.6. Exercise s 8 5

Chapter 7 . Sum s o f Square s 8 7 7.1. Positiv e Semidefinit e Matrice s 8 7 7.2. Zero-dimensiona l Ideal s an d SOStool s 8 9 7.3. Globa l Optimizatio n 9 2 7.4. Th e Rea l Nullstellensat z 9 4 7.5. Symmetri c Matrice s wit h Doubl e Eigenvalue s 9 6 7.6. Exercise s 9 9

Chapter 8 . Polynomia l System s i n Statistic s 10 1 8.1. Conditiona l Independenc e 10 1 8.2. Graphica l Model s 10 4 8.3. Rando m Walk s o n th e Intege r Lattic e 10 9 8.4. Maximu m Likelihoo d Equation s 11 4 8.5. Exercise s 11 7

Chapter 9 . Tropica l Algebrai c Geometr y 11 9 9.1. Tropica l Geometr y i n the Plan e 11 9 9.2. Amoeba s an d thei r Tentacle s 12 3 9.3. Th e Bergma n Comple x o f a Linea r Spac e 12 7 9.4. Th e Tropica l Variet y o f an Idea l 12 9 9.5. Exercise s 13 1

Chapter 10 . Linea r Partia l Differentia l Equation s with Constan t Coefficient s 13 3

10.1. Wh y Differentia l Equations ? 13 3 10.2. Zero-dimensiona l Ideal s 13 5 10.3. Computin g Polynomia l Solution s 13 7 10.4. Ho w to Solv e Monomia l Equation s 14 0 10.5. Th e Ehrenpreis-Palamodo v Theore m 14 1 10.6. Noetheria n Operator s 14 2 10.7. Exercise s 14 4

Bibliography 14 7

Index 151

Preface

This boo k gre w ou t o f th e note s fo r te n lecture s give n b y th e autho r a t th e CBMS Conferenc e a t Texa s A& M University , Colleg e Station , durin g th e wee k o f May 20-24, 2002. Paul o Lima Filho, J. Maurice Rojas an d Hal Schenck did a fantas -tic job o f organizing thi s conferenc e an d takin g car e o f mor e tha n 8 0 participants , many o f the m graduat e student s workin g i n a wid e rang e o f mathematica l fields . We were fortunate t o be abl e to liste n t o th e excellen t invite d lecture s delivere d b y the following twelve leading experts: Saugat a Basu , Eduard o Cattani , Karin Gater -mann, Crai g Huneke , Tien-Yie n Li , Gregori o Malajovich , Pabl o Parrilo* , Mauric e Rojas, Fran k Sottile , Mik e Stillman* , Thorste n Theobald , an d Ja n Verschelde* .

Systems o f polynomia l equation s ar e fo r everyone : fro m graduat e student s in compute r science , engineering , o r economic s t o expert s i n algebrai c geometry . This boo k aim s t o provid e a bridg e betwee n mathematica l level s an d t o expos e a s many facet s o f the subjec t a s possible . I t cover s a wide spectru m o f mathematica l techniques an d algorithms , bot h symboli c an d numerical . Ther e ar e tw o chapter s on applications . Th e on e abou t statistic s i s motivate d b y th e author' s curren t research interests , an d th e on e abou t economic s (Nas h equilibria ) recognize s Dav e Bayer's rol e i n th e makin g o f th e movi e A Beautiful Mind. (Man y thanks , Dave , for introducin g m e to th e star s a t thei r kick-of f part y i n NY C o n Marc h 16 , 2001. )

At th e en d o f eac h chapte r ther e ar e abou t te n exercises . Thes e exercise s vary greatl y i n thei r difficulty . Som e ar e straightforwar d application s o f materia l presented in the text while other "exercises " ar e quite hard and ought to be renamed "suggested researc h directions" . Th e reade r ma y decid e whic h i s which .

We ha d a n inspirin g softwar e sessio n a t th e CBM S conference , an d th e jo y of computin g i s reflecte d i n thi s boo k a s well . Sprinkle d throughou t th e text , the reade r find s shor t compute r session s involvin g polynomia l equations . Thes e involve the commercial packages Maple® an d MATLAB® as well as the freely availabl e packages Singular 1 , Macaula y 2 2, PHCpack 3, an d SOSTools 4. Developer s o f th e last thre e program s spok e a t th e CBM S conference . Thei r name s ar e marke d wit h a sta r above .

There ar e man y fin e compute r program s fo r solvin g polynomia l system s othe r than th e one s liste d above . Sadly , I di d no t hav e tim e t o discus s the m all . On e

1 Singular i s a free softwar e distribute d unde r th e GN U license . © Departmen t o f Mathematic s and Centr e fo r Compute r Algebra , Universit y o f Kaiserslautern , German y

2Macaulay2: ©Danie l R . Grayso n an d Michae l E . Stillma n (1993-2001 ) an d i s distribute d free unde r th e GN U licens e

3PHCpack: © 1998 , Katholieke Universitei t Leuven , Departmen t o f Compute r Science , Hev -erlee, Belgiu m

4SOSTools i s a MATLAB ® toolbo x an d freel y availabl e unde r th e GN U licens e at : http://www.cds.caltech.edu/sostools o r http://www.aut.ee.ethz.ch/~parrilo/sostool s

vii

viii PREFAC E

such progra m i s CoCoA 5 whic h i s comparabl e t o Singula r an d Macaula y 2 . Th e textbook b y Kreuze r an d Robbian o [KROO ] doe s a wonderfu l jo b introducin g th e basics o f Computationa l Commutativ e Algebr a togethe r wit h example s i n CoCoA.

Software i s necessarily ephemeral. Whil e the mathematics of solving polynomial systems continue s t o liv e fo r centuries , th e compute r cod e presente d i n thi s boo k will becom e obsolet e muc h sooner . I teste d i t al l i n Ma y 2002 , an d i t worke d wel l at tha t time , o n ou r departmenta l compute r syste m a t U C Berkeley . An d i f yo u would lik e to find ou t more , eac h o f these program s ha s excellen t documentation .

I a m gratefu l t o th e student s i n m y graduat e course , Math 275: Topics in Ap-plied Mathematics, fo r listenin g t o my te n lecture s a t hom e i n Berkele y whil e I first assemble d the m i n th e sprin g o f 2002 . Thei r spontaneou s comment s prove d to b e extremel y valuabl e fo r improvin g m y performanc e late r o n i n Texas . Afte r the CBM S conference , th e followin g peopl e provide d ver y helpfu l comment s o n my manuscript: Joh n Dalbec , Jesu s D e Loera , Mik e Develin , Alici a Dickenstein , Ia n Dinwoodie, Bahma n Engheta , Stephe n Fulling , Kari n Gatermann , Raymon d Hem -mecke, Serkan Ho§ten, Robert Lewis , Gregorio Malajovich, Pabl o Parrilo, Francisc o Santos, Fran k Sottile , Set h Sullivant , Cale b Walther , an d Dongshen g Wu .

Special thank s g o t o Ami t Kheta n an d Ruchir a Datt a fo r helpin g m e whil e in Texa s an d fo r contributin g t o Section s 4. 5 an d 6. 2 respectively . Ruchir a als o assisted m e in the har d wor k o f preparing th e final versio n o f this book . I t wa s he r help tha t mad e th e rapi d completio n o f thi s projec t possible .

Last bu t no t least , I wis h t o dedicat e thi s boo k t o th e bes t tea m o f all : m y daughter Nina , my son Pascal , an d my wife Hyungsook . A million thanks fo r bein g patient wit h you r pap a an d puttin g u p wit h hi s craz y early-mornin g wor k hours .

Bernd Sturmfels * Berkeley, Jun e 200 2

5 A. Capani , G . Niesi , L. Robbiano, CoCoA , a system fo r doing Computations i n Commutativ e Algebra, availabl e vi a anonymou s ft p from : http://cocoa.dima.unige.i t

*The autho r wa s supporte d i n par t b y th e U.S . Nationa l Scienc e Foundation , grant s #DMS -0200729 an d #DMS-013832 3

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Index

algebraic function , 8 , 3 3 algebraic torus , 31 , 12 3 Alon-Jaeger-Tarsi conjecture , 6 9 amoebas, 11 9 Artin's Theorem , 9 5

Bezout matrix , 44 , 4 8 Bezout's Theorem , 29 , 12 3 Bergman complex , 12 4 Bernstein's Theorem , 31 , 37, 12 3 binary rando m variable , 103 , 10 7 binomial, 32 , 62 , 10 4 Birch's Theorem , 11 4 bisection, 5

Castelnuovo-Mumford regularity , 7 0 Cholesky decomposition , 8 8 circuit ideal , 113 , 14 3 companion matrix , 4 , 11 , 17-20 , 22 , 26 , 5 7

resultant formula , 4 4 complete intersection , 29 , 63 , 67 , 107 , 13 1 cubic surface , 2 6

derangement, 8 2 Descartes's Rul e o f Signs , 5 discriminant, 10 , 11 , 23, 135 , 13 9

computation of , 1-2 , 29-3 0 of symmetri c matrix , 96-9 9

Ehrenpreis-Palamodov Theorem , 133 , 141 , 142

eigenvalues, 4 , 11 , 17-22 , 87-88 , 96-9 9 elimination ideal , 14-1 5 elimination theory , 49 , 51 , 52 elliptic curve , 27 , 12 2 entire function , 134 , 13 6

fewnomial, 3 8 Fibonacci number , 6 4 floating point , 3 , 87 , 8 8

Galois group , 2-3 , 2 6 game i n norma l form , 7 1 Gel'fand-Kapranov-Zelevinsky, 10 , 50 , 12 4 Gorenstein, 135 , 139-14 0 Grobner duality , 13 5 graphical model , 82 , 101 , 103 , 10 4 Grassmannian, 127 , 13 1

Grobner basis , 29 , 90 , 93 , 105 , 107 , 127 and Bergma n fan , 12 5 computation, compare d t o Hermit e nor -

mal form , 3 3 implicitizing rationa l plan e curv e using ,

45 of adjacen t minors , 6 4 of decomposabl e graph , 10 4 of lattic e walks , 11 1 of subpermanents , 67 , 6 8 of totall y mixe d Nas h equilibria , 7 4 of zero-dimensiona l ideal , 13-27 , 89 , 9 0 statistics and , 10 1 universal, 12 7

Grobner cone , 125-12 7 Grobner fan , 125-12 7

Hammersley-Clifford Theorem , 10 4 harmonic polynomial , 13 9 Hermite norma l form , 3 2 Hilbert series , 9 8 homotopy method , 37 , 8 2 hypergeometric differentia l equations , 1 0 hypergeometric function , 9

ideal quotient , 15-17 , 13 7 Ilyushechkin's Theorem , 9 8 initial form , 12 5 initial ideal , 125-127 , 12 9 initial monomial , 5 2 initial monomia l ideal , 13 , 60 , 12 7 integral representation , 14 4 irreducible variety , 59 , 103 , 108 , 125 , 14 1 iterative proportiona l scaling , 11 6

Jordan canonica l form , 1 1

Khovanskii's Theorem , 3 8

Laurent polynomial , 22 , 24 , 31 , 49, 52 , 12 4

Laurent series , 6 lexicographic ter m order , 15 , 26 , 4 4 linear programming , 17 , 89 , 9 4 local ring , 16 , 17 , 13 5 lower facet , 3 5

Macaulay 2 , 14 , 5 9 Mathieu's problem , 24 , 2 7

152 INDEX

matroid, 12 7 maximal ideal , 15 , 17 , 136 , 137 maximum likelihood , 11 5 Minkowski sum , 31 , 35, 80 , 12 3 mixed volume , 37 , 50 , 73 , 80 , 8 4 moment map , 11 5 monomial ideal , 62 , 14 0 multiplicity, 1 , 6 , 15 , 16 , 133-13 6

not i n trac e form , 2 0 standard monomial s and , 13 , 1 6 tropical, 12 3

Nash equilibrium , 7 1 Newton polytope , 30-37 , 51 , 52, 55 , 73 , 79 -

81 amoeba and , 12 4 Bergman fa n and , 12 5 Puiseux serie s and , 7 tropical curve s with , 12 3

nodal curve , 27 , 13 1 non-radical idea l

scheme structur e of , 13 4 normal form , 14 , 17 , 89 , 13 8

optimization, 9 1

partial differentia l equations , 8 , 16 , 13 3 payoff matrix , 7 1 permanent, 67 , 68 , 83 , 14 5 Petersen graph , 12 7 PHCpack, 8 2 piecewise-linear, 7 , 34-36 , 11 9 polytope, 3 0 positive semidefinite , 88 , 9 2 primary decomposition , 17 , 26 , 59-7 0

Markov chain s and , 10 1 of binomia l ideal , 62 , 10 4 of circui t ideals , 113 , 11 4 of graphica l models , 103 , 10 4 of idea l o f partia l differentia l operators ,

136, 137 , 14 2 random walk s and , 111 , 112 , 11 7

prime ideal , 15 , 18 , 59, 60 , 65, 102-104 , 125 , 144

Puiseux series , 6-11 , 34-35 , 37 , 4 1 valuation and , 119 , 13 0

quadratic form , 20 , 48 , 87 , 88 , 90 , 13 9

radical ideal , 13-15 , 17-18 , 26 , 59 , 7 4 and associate d primes , 60-6 1 and primar y decomposition , 6 1 and primar y ideal , 17 , 62 , 13 6 close t o zero-dimensiona l ideal , 2 6 computing i n Macaulay2 , 6 8 Ehrenpreis-Palamodov pair s of , 14 2 in univariat e polynomia l ring , 4- 5 maximizing polynomia l modulo , 9 2 maximum likelihoo d idea l as , 11 5 of adjacen t minors , 64-6 6

of embedde d component , 6 6 primary decomposition , 6 6

of binomia l ideal , 6 2 of circui t ideal , 11 3 of graphica l model , 103 , 10 5 of independenc e model , 10 6 of subpermanents , 67-68 , 7 0 of univariat e polynomial , 4 zero-dimensional, 1 4

rational univariat e representation , 1 5 real Nullstellensatz , 9 4 regular subdivision , 12 1 resultant, 2 , 30 , 43-5 7

companion matri x formula , 4 4

saturated independenc e condition , 104-11 4 saturation, 15-17 , 106 , 12 5 scheme, 16 , 134 , 14 2 semidefinite programming , 87 , 88 , 9 0 shape lemma , 1 5 sign changes , 5-6 , 2 1

in characteristi c polynomial , 2 2 Singular, 15 , 23, 5 9 solving th e quintic , 9 SOStools, 8 9 sparse polynomial , 3 , 29-41 , 49 , 79 , 8 0 sparse resultant , 49-5 7 squarefree, 4 , 14 , 6 0 standard monomials , 13 , 14 , 89, 90 , 137-14 0

in resultan t computation , 5 5 multiplicity and , 1 6

standard pairs , 140 , 14 2 state polytope , 12 5 strategy, 71 , 74 Sturm sequence , 5 sufficient statistic , 11 4 Sylvester matrix , 2 , 44 , 48 , 5 5 symbolic power , 14 4

toric deformation , 3 3 toric ideal , 104 , 10 6 toric Jacobian , 4 0 trilinear equation , 55 , 7 6 tropical curve , 12 0 tropical semiring , 11 9 tropical variety , 13 0

unmixed, 38 , 52 , 5 5

zero-dimensional ideal , 13-27 , 47 , 59 , 89-9 2 complete intersection , 6 3 Gorenstein, 13 9 maximum likelihoo d ideal , 11 5 monomial, 14 0 of partia l differentia l operators , 136 , 137 ,

141, 14 2