Libraries and resellers, please contact cust-serv ams. See our librarian page for additional eBook ordering options. 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. These applications lie in the domains of integer programming and computational statistics.
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Among them, one of the most important topics is the correspondence to triangulations of convex polytopes. The purpose of this chapter is to explain these topics in detail.
First, we will explain convex polytopes, weight vectors, and monomial orders, all of which play a basic role in the rest of this chapter. Third, we will discuss the correspondence between the initial ideals of toric ideals and triangulations of convex polytopes, and the related ring-theoretic properties.
Advertisement Hide. Chapter First Online: 03 December This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Bruns, R. Hemmecke, B.
Ichim, M. Gelfand, M. Kapranov, A. Zelevinski, Discriminants, Resultants, and Multidimensional Determinants. Ohsugi, T. Hibi, Normal polytopes arising from finite graphs. Hibi, A normal 0,1 -polytope none of whose regular triangulations is unimodular. Discrete Comput. Hibi, Toric ideals arising from contingency tables, in Commutative Algebra and Combinatorics , ed. Ohsugi, J. Herzog, T. Hibi, Combinatorial pure subrings. Osaka J. Saito, B.
Sturmfels, N. Algorithms and Computation in Mathematics, vol. Simis, W. Vasconcelos, R. Villarreal, The integral closure of subrings associated to graphs. Sullivant, Compressed polytopes and statistical disclosure limitation. Tohoku Math. Thomas, Lectures in Geometric Combinatorics. Personalised recommendations.
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Convex Polytopes and Gröbner Bases
Grobner Bases and Convex Polytopes. Bernd Sturmfels. 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.
Gröbner Bases and Convex Polytopes