Grobner Bases and Convex Polytopes

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American Mathematical Soc., 1996 - Mathematics - 162 pages
This book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centers around a special class of ideals in a polynomial ring: the class of toric ideals. They are characterized as those prime ideals that are generated by monomial differences or as the defining ideals of toric varieties (not necessarily normal).
The interdisciplinary nature of the study of Grobner bases is reflected by the specific applications appearing in this book. These applications lie in the domains of integer programming and computational statistics. The mathematical tools presented in the volume are drawn from commutative algebra, combinatorics, and polyhedral geometry.
 

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Contents

Chapter 1 Gröbner Basics
1
Chapter 2 The State Poly tope
9
Chapter 3 Variation of Term Orders
19
Chapter 4 Toric Ideals
31
Chapter 5 Enumeration Sampling and Integer Programming
39
Chapter 6 Primitive Partition Identities
47
Chapter 7 Universal Grobner Bases
55
Chapter 8 Regular Triangulations
63
Chapter 10 Agraded Algebras
85
Chapter 11 Canonical Subalgebra Bases
99
Chapter 12 Generators Betti Numbers and Localizations
113
Chapter 13 Toric Varieties in Algebraic Geometry
127
Chapter 14 Some Specific Grobner Bases
141
Bibliography
155
Index
161
Back Cover
163

Chapter 9 The Second Hypersimplex
75

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