Valuations on lattice polytopes

This survey is on classification results for valuations defined on lattice polytopes that intertwine the special linear group over the integers. The basic real valued valuations, the coefficients of the Ehrhart polynomial, are introduced and their characterization by Betke and Kneser is discussed. More recent results include classification theorems for vector and convex body valued valuations. © Springer International Publishing AG 2017

Lattice polytopes having h^*-polynomials with given degree and linear coefficient

The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*_1. We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope, if the dimension of P is greater or equal to h^*_1 (2d+1) + 4d-1. This result has a purely combinatorial proof and generalizes a recent theorem of Batyrev.Comment: AMS-LaTeX, 9 pages; introduction improve

Periods of Ehrhart Coefficients of Rational Polytopes

Let P⊂R^n be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the kth dilate of P (k a positive integer) is a quasi-polynomial function of k — that is, a "polynomial" in which the coefficients are themselves periodic functions of k. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values

Difference of Hilbert series of homogeneous monoid algebras and their normalizations

The version of record of this article, first published in Semigroup Forum, is available online at Publisher’s website: https://doi.org/10.1007/s00233-024-10414-0.Let Q be an affine monoid, k[Q] the associated monoid k-algebra, and k[Q¯] its normalization, where we let k be a field. We discuss a difference of the Hilbert series of k[Q] and k[Q¯] in the case where k[Q] is homogeneous (i.e., standard graded). More precisely, we prove that if k[Q] satisfies Serre’s condition (S₂), then the degree of the h-polynomial of k[Q] is always greater than or equal to that of k[Q¯]. Moreover, we also show counterexamples of this statement if we drop the assumption (S₂)