h-vectors of Gorenstein polytopes

Details h-vectors of Gorenstein polytopes h-vectors of Gorenstein polytopes

2005-08-21

arxiv.org/abs/math/0508392v2

J. Comb. Theory, Ser. A 114, No. 1, 65-76 (2007)

Mathematics

Commutative Algebra

12 pages; besides minor modifications the main result needs the additional assumption that the polytope P has a regular unimod

Scientific paper

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 in N, can be written as the sum of n 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.

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