Other Title s in This Serie s

Volume 8 Bern d Sturmfel s

Grobner base s an d conve x polytope s 1996

7 And y R . Magi d Lectures o n differentia l Galoi s theor y 1994

6 Dus a McDuf f an d Dietma r Salamo n J-holomorphic curve s an d quantu m cohomolog y 1994

5 V . I . Arnol d Topological invariant s o f plan e curve s an d caustic s 1994

4 Davi d M . Goldschmid t Group characters , symmetri c functions , an d th e Heck e algebr a 1993

3 A . N . Varchenk o an d P . I . Etingo f Why th e boundar y o f a roun d dro p become s a curv e o f orde r fou r 1992

2 Frit z Joh n Nonlinear wav e equations , formatio n o f singularitie s 1990

1 Michae l H . Freedma n an d Fen g Lu o Selected application s o f geometr y t o low-dimensiona l topolog y 1989

http://dx.doi.org/10.1090/ulect/008

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Grobner Base s and Conve x Polytope s

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University

LECTURE Series

Volume 8

Grobner Base s and Conve x Polytope s

Bernd Sturmfel s

American Mathematica l Societ y Providence, Rhod e Islan d

Editorial Committe e Jerry L . Bon a Donal d S . Ornstei n Theodore W . Gameli n Leonar d L . Scot t (Chair )

1991 Mathematics Subject Classification. P r imar y 13P10 , 14M25 ; Secondary 52B12 , 90C10 , 14Q99 .

ABSTRACT. Thi s boo k i s about th e interpla y o f computational commutativ e algebr a an d th e theor y of conve x polytopes . A centra l them e i s th e stud y o f tori c ideal s an d thei r application s i n intege r programming. Thi s boo k i s aime d a t graduat e student s i n mathematics , compute r science , an d theoretical operation s research .

Library o f Congres s Cataloging-in-Publicat io n Dat a

Sturmfels, Bernd , 1962 -Grobner base s an d conve x polytope s / Bern d Sturmfels .

p. cm . — (Universit y lectur e series , ISS N 1047-3998 ; v . 8 ) Includes bibliographica l reference s an d index . ISBN 0-8218-0487- 1 1. Grobne r bases . 2 . Conve x polytopes . I . Title . II . Series : Universit y lectur e serie s

(Providence, R.I.) ; 8 . QA251.3.S785 199 5 512/.24—dc20 95-4578 0

CIP

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except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

U Printe d o n recycle d paper .

10 9 8 7 6 5 4 3 2 0 1 0 0 9 9 9 8 9 7

Contents

Introduction i x

Chapter 1 . Grobner Basic s 1

Chapter 2 . The Stat e Pol y tope 9

Chapter 3 . Variation o f Term Order s 1 9

Chapter 4 . Toric Ideal s 3 1

Chapter 5 . Enumeration , Samplin g an d Intege r Programmin g 3 9

Chapter 6 . Primitiv e Partitio n Identitie s 4 7

Chapter 7 . Universa l Grobne r Base s 5 5

Chapter 8 . Regula r Triangulation s 6 3

Chapter 9 . Th e Secon d Hypersimple x 7 5

Chapter 10 . ^4-graded Algebra s 8 5

Chapter 11 . Canonical Subalgebr a Base s 9 9

Chapter 12 . Generators , Bett i Number s an d Localization s 11 3

Chapter 13 . Toric Varietie s i n Algebrai c Geometr y 12 7

Chapter 14 . Some Specifi c Grobne r Base s 14 1

Bibliography 15 5

Index 16 1

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Introduction

Grobner base s theory provide s th e foundatio n fo r man y algorithm s i n algebrai c ge-ometry an d commutativ e algebra , wit h th e Buchberge r algorith m actin g a s the en -gine that drive s the computations. Thank s to the text book s by Adams-Loustauna u (1994), Becker-Weispfenning (1993) , Cox-Little-O'Shea (1992 ) and Eisenbud (1995) , Grobner base s ar e now entering the standar d algebr a curriculu m a t man y universi -ties. I n view of the ubiquity of scientific problem s modeled by polynomial equations , this subjec t i s o f interes t no t onl y t o mathematicians , bu t als o t o a n increasin g number o f scientist s an d engineers .

The interdisciplinar y natur e o f th e stud y o f Grobne r base s i s reflected b y th e specific application s appearin g i n thi s book . Thes e application s li e in th e domain s of intege r programmin g an d computationa l statistics . Th e mathematica l tool s t o be presente d ar e draw n fro m commutativ e algebra , combinatorics , an d polyhedra l geometry.

The mai n threa d o f thi s boo k center s aroun d a specia l clas s o f ideal s i n a polynomial ring , namely , th e clas s o f toric ideals. The y ar e characterize d a s thos e prime ideal s that ar e generated b y monomial differences , o r a s the defining ideal s of (not necessaril y normal ) tori c varieties . Tori c ideals are intimately relate d t o recen t advances in polyhedral geometry , which grew out o f the theory o f *4-hypergeometri c functions du e t o Gel'fand , Kaprano v an d Zelevinsk y (1994) . A ke y concep t i s that o f a regular triangulation. Al l regula r triangulation s o f a fixed polytop e ar e parametrized b y th e vertice s o f the secondary polytope.

Both the algebra and the combinatorics appearin g in this book are presented a s self-contained a s possible , Mos t o f th e materia l i s accessible t o first-year graduat e students i n mathematics . Th e followin g prerequisite s will b e assume d throughout :

• workin g knowledg e o f th e basi c fact s abou t Grobne r bases , specificall y o f Chapters 1- 5 an d 9 of (Cox-Little-O'She a 1992) , or Chapter s 1- 2 o f (Adams -Loustaunau 1994) ,

• familiarit y wit h th e terminolog y o f polyhedra l geometr y an d linea r program -ming, a s introduce d i n (Schrijve r 1986 ) o r (Ziegle r 1995) .

The fourtee n chapter s ar e organize d a s follows . I n th e first tw o chapter s we presen t som e introductor y Grobne r base s material , whic h canno t b e foun d i n the tex t books . Her e w e conside r arbitrar y ideal s I i n a polynomia l rin g S — k[x\,..., # n] , no t jus t tori c ideals . I t i s prove d tha t / admit s a universal Grobner basis, tha t is , a finite subse t whic h i s a Grobne r basi s fo r I wit h respec t t o al l term order s simultaneously . Thi s lead s t o th e concep t o f th e Grobner fan an d th e state polytope o f / . Th e stat e polytop e i s a conve x polytop e i n R n whos e vertice s are i n bijectio n wit h th e distinc t initia l monomia l ideal s o f / wit h respec t t o al l term order s o n S. I n th e specia l cas e wher e I = (/ ) i s a principa l ideal , th e stat e

x I N T R O D U C T I O N

polytope coincide s wit h th e familia r Newto n polytop e New(f). I n Chapte r 3 we present result s an d algorithm s whic h involve the variation o f term order s fo r a fixe d ideal. I n particular , i t i s explained ho w to comput e th e stat e polytope .

Chapters 4- 9 dea l exclusivel y wit h tori c ideals . Basi c algebrai c feature s o f these ideal s ar e collecte d i n th e fourt h chapter . Thes e includ e degre e bound s fo r Grobner bases . A n explici t universa l Grobne r basis , th e so-calle d Graver basis, is constructed.

The fift h chapte r relate s toric ideal s to three fundamenta l problem s associate d with a (non-negativ e integer ) linea r map IT : Nn — > N d . Eac h fiber 7r -1(b), b € N d , consists o f th e lattic e point s i n a polyhedro n i n R n . Th e thre e problem s are : enumeration (lis t al l points in 7r_1 (b)), sampling (pic k a point i n 7r_1(b) a t random ) and integer programming (fin d a poin t i n 7r _1(b) whic h minimize s a give n linea r functional). Th e tori c idea l I A whic h i s used t o mode l thes e question s i s the idea l of algebrai c relation s o n th e monomial s wit h exponen t vector s th e column s o f a matrix A representin g 7r .

Chapter 6 deals wit h th e one-dimensiona l cas e d = 1 . I n thi s cas e th e Grave r basis elements of I A correspon d t o primitive partition identities, th e variety define d by I A i s a monomia l curve , an d th e associate d intege r progra m i s th e knapsack problem. Shar p degre e bound s fo r Grobne r base s ar e availabl e i n thi s case .

Returning t o our genera l discussion i n Chapte r 7 , we present a geometric char -acterization o f the universa l Grobne r basi s UA, &n d we discuss algorithm s fo r com -puting UA> I n Chapte r 8 we establish a correspondence betwee n the initia l ideal s of I A an d th e regula r triangulation s o f A. Thes e triangulation s ar e parametrize d b y the secondary polytop e T,(A), whic h i s a Minkowski summan d o f the stat e polytop e of I A- I n Chapte r 9 we apply ou r genera l theor y t o a specifi c famil y o f toric ideals , namely thos e define d b y th e second hypersimplex.

In th e nex t tw o chapter s w e ventur e int o th e real m o f commutativ e alge -bra beyon d tori c ideals . Chapte r 1 0 deal s wit h Arnold' s notio n o f A-graded al-gebras. Thes e ar e algebra s wit h th e simples t possibl e Hilber t function , namely , 1,1,1,1,1,1, . . . . Thei r definin g ideal s ar e a natura l generalizatio n o f initia l ideal s of tori c ideals . I n Chapte r 1 1 w e discus s canonical subalgebra bases (o r SAGBI bases, a s the y wer e calle d b y Robbian o & Sweedle r (1990)) . Thes e base s nee d not b e finite. Bu t i f the y are , the n the y admi t a nic e geometri c interpretatio n a s degeneration o f a parametrically presente d variet y int o a tori c variety .

In Chapte r 1 2 we present advance d technique s fo r computin g wit h tori c ideals , and fo r applyin g the m t o intege r programming . Chapte r 1 3 aims t o spa n a bridg e to th e theor y o f tori c varietie s a s i t exist s i n algebrai c geometry , and , finally, i n Chapter 1 4 the reader finds a collection o f toric ideal s an d Grobne r base s which ar e dear t o th e author' s heart .

At th e en d o f eac h chapte r ther e i s a lis t o f exercise s an d bibliographi c notes . The exercise s var y i n difficulty : som e ar e straightforwar d application s o f th e ma -terial presented , whil e other s ar e mor e difficul t an d ma y lea d t o researc h projects . Many o f the exercises assume a certain leve l of enthusiasm fo r performin g compute r experiments. Th e bibliographi c note s ar e kep t ver y brief . Thei r mai n purpos e i s to hel p th e reade r i n locatin g a sampl e o f th e origina l o r backgroun d literature . They ar e no t intende d t o giv e a historica l accoun t o r a complet e bibliograph y o f the respectiv e subjec t areas .

I N T R O D U C T I O N x i

This monograp h gre w ou t o f a serie s o f ten lecture s give n a t th e Holida y Sym -posium a t Ne w Mexic o Stat e University , La s Cruces , Decembe r 27-31 , 1994 . Th e material wa s update d an d expande d considerabl y afte r th e Holida y Symposium . In particular , Chapter s 3 , 12 , 1 3 an d 1 4 wer e added . I a m gratefu l t o numerou s participants wh o supplie d comment s an d helpe d m e i n locatin g error s i n previou s versions. Serka n Hoste n di d a particularly grea t job durin g the las t roun d o f proof-reading. I wis h t o than k al l my co-author s liste d i n th e bibliograph y fo r inspirin g collaborations. Man y o f the technique s an d result s presente d i n thi s book I learne d from an d wit h them .

This projec t wa s supported financially b y the Davi d an d Lucil e Packard Foun -dation an d th e Nationa l Scienc e Foundatio n throug h a n NY I Fellowship . Mos t o f the writin g wa s don e durin g m y 94/9 5 visi t a t th e Couran t Institut e o f Ne w Yor k University. I wish t o than k m y host s a t NY U an d a t Ne w Mexic o Stat e Universit y for thei r terrifi c hospitality .

Special thanks g o to Hyungsoo k an d Nina , fo r thei r support , lov e and cheerfu l energy.

Bernd Sturmfel s Berkeley, Augus t 199 5

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Index

„4-graded algebra , 8 5

Betti number , 12 0 Birkhoff pol y tope, 14 5 Buchberger algorithm , 1 , 10 2

canonical basis , 9 9 circuit, 3 , 33 , 81 Cohen-Macaulay, 13 5 coherent, 8 5 complex, 1 1 conformal, 3 4 contingency table , 4 0

diagonal ter m order , 4 3

Ehrhart polynomial , 36 , 79 , 133 , 135

face, 1 0 fiber, 3 9

Grassmann variety , 29 , 103 , 110 , 13 6 Graver basis , 35 , 48, 56 , 81 , 150 Graver degree , 59 , 87 Grobner basi s detection , 2 5 Grobner degree , 59 , 89 Grobner fan , 13 , 71 Grobner fiber, 6 0 Grobner region , 5

Hilbert basis , 12 8 Hilbert polynomial , 36 , 79 , 13 3 hypersimplex, 75 , 84

ideal quotient , 11 3 initial algebra , 9 9 initial complex , 63 , 102 initial ideal , 1 , 4, 85 , 10 6 integer programming , 39 , 43 , 88, 12 2

knapsack problem , 5 2

Lawrence lifting , 55 , 149 lexicographic (ter m order /

triangulation), 1 , 67 , 14 3 lineality space , 1 9 linear programming , 26 , 65

MACAULAY, 86 , 89 , 15 1 MAPLE, 8 9 marked polynomial , 2 6 matroid pol y tope, 1 6 minimal Grobne r basis , 1 Minkowski addition/sum , 10 , 61 mono-AGA, 86 , 9 4

Newton polytope , 11 , 19 normal (tori c variety) , 127 , 130 , 13 4 normal cone , 1 1 normal fan , 1 1 normalized volume , 3 6

permutation matrix , 14 5 primary decomposition , 12 2 primitive, 33 , 47, 87 projectively normal , 132 , 13 6 polyhedral subdivision , 9 0 polytope, 9

random walk , 4 1 reduced Grobne r basis , 1 , 32 , 76 reduced lattic e basis , 11 5 regular (triangulation) , 6 4 reverse lexicographi c (ter m order /

triangulation), 1 , 113 , 143 , 147

sampling, 39 , 84 secondary polytope , 71 , 79 Segre (variety) , 37 , 14 9 standard (monomial) , 1 , 86, 88

161

162 INDEX

state polytope , 14 , 15 , 19 , 61, 71, 79, 10 2 syzygy, 107 , 108 , 120

tangent cone , 13 0 term order , 1 , 5 thrackle, 7 9 three-dimensional matrix , 14 8 transportation problem , 4 0 triangulation, 63 , 123 truncated Grobne r basis , 11 8 toric ideal , 31 , 100 toric variety , 31 , 36, 127 , 12 9

unimodular, 69 , 70 , 93, 13 6 universal Grobne r basis , 2 , 6 , 15 , 33, 57, 81

Veronese, 14 1