Ehrhart Theory of Spanning Lattice Polytopes

Abstract

The key object in the Ehrhart theory of lattice polytopes is the numerator polynomial of the rational generating series of the Ehrhart polynomial, called |$h^*$|-polynomial. In this article, we prove a new result on the vanishing of its coefficients. As a consequence, we get that |$h^*_i =0$| implies |$h^*_{i+1}=0$| if the lattice points of the lattice polytope affinely span the ambient lattice. This generalizes a recent result in algebraic geometry due to Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud–Goto conjecture. We also discuss how this study is motivated by unimodality questions and how it relates to decomposition results on lattice polytopes of given degree. The proof methods involve a novel combination of successive modifications of half-open triangulations and considerations of number-theoretic step functions.