Inequalities and Ehrhart $\delta$-Vectors

Abstract

For any lattice polytope $P$, we consider an associated polynomial ${\bar{\delta }}_P(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\delta$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope.

11 pages. v2: minor changes, more detailed proof of Lemma 2.12. To appear in Trans. Amer. Math. Soc