h-Vectors of Gorenstein polytopes

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:114

Article

Bruns, Winfried ; Roemer, Tim

关键词: lattice polytope;    h-Vector;    Ehrhart function;    Gorenstein ring;    affine monoid;    unimodality;    triangulation;    initial ideal;   

DOI  :  10.1016/j.jcta.2006.03.003

来源: Elsevier

【 摘 要 】

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K [M] (K a field) by a long regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, C is an element of N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies. (c) 2006 Elsevier Inc. All rights reserved.

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