Introduction Grobner base s theory provide s the 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 problems 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 scientists 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 ideals that ar e generated b y monomial differences, o r as the defining ideal s of (not necessaril y normal) tori c varieties. Tori c ideals are intimately relate d t o recent advances in polyhedral geometry, which grew out of the theory of *4-hypergeometric 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 appearing 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 ], not 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
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