Ehrhart Theory of Spanning Lattice Polytopes

TL;DR

This paper proves that spanning lattice polytopes have no gaps in their $h^*$-vector, connecting Ehrhart theory with algebraic and combinatorial properties, and discusses implications for unimodality and polyhedral decompositions.

Contribution

It establishes that spanning lattice polytopes' $h^*$-vectors are gapless, generalizing recent results and linking to the Eisenbud-Goto conjecture.

Findings

$h^*$-vector of spanning lattice polytopes has no gaps

Generalizes a recent result by Blekherman, Smith, and Velasco

Connects Ehrhart theory to algebraic geometry and unimodality questions

Abstract

A lattice polytope is called spanning if its lattice points affinely span the ambient lattice. We show as a corollary to a general result in the Ehrhart theory of lattice polytopes that the $h^*$-vector of a spanning lattice polytope has no gaps, i. e., $h^*_i =0$ implies $h^*_{i+1}=0$. This generalizes a recent result by Blekherman, Smith, and Velasco, and implies a polyhedral consequence of the Eisenbud-Goto conjecture. We also discuss how this relates to unimodality questions of lattice polytopes and previously achieved decomposition results on lattice polytopes of given degree.