Boundary $h^\ast$-vectors and unimodular triangulations

TL;DR

This paper explores the relationship between Ehrhart $h^ extast$-polynomials of lattice polytope boundaries and regular unimodular triangulations, establishing new structural and characterization results.

Contribution

It introduces a boundary analogue of Sturmfels correspondence, linking Ehrhart $h^ extast$-polynomials to triangulation $h$-polynomials, and derives new symmetry and unimodality results.

Findings

Established a boundary analogue of Sturmfels correspondence.

Connected Ehrhart $h^ extast$-polynomials to face enumeration of simplicial complexes.

Derived Dehn-Sommerville-type relations and bounds for $h^ extast$-coefficients.

Abstract

We study the Ehrhart $h^\ast$-polynomial of (the boundary of) a lattice polytope via regular unimodular triangulations and Gr\"obner degenerations of toric ideals. Our main result is a boundary analogue of the well-known Sturmfels correspondence. This allows us to connect the boundary $h^\ast$-polynomial to the $h$-polynomial of any regular unimodular triangulation, in analogy to the classical Betke-McMullen Theorem. Providing a direct link between Ehrhart theory and the face enumeration of simplicial complexes, we then transfer structural results from the theory of simplicial polytopes to the setting of lattice polytopes. In particular, we derive general Dehn-Sommerville-type relations between $h^\ast(P)$ and $h^\ast(\partial P)$. Under the additional assumption of $\partial P$ admitting a regular unimodular triangulation, we recover old and prove new characterization results concerning symmetry or unimodality, as well as upper and lower bounds for coefficient-wise differences within $h^\ast(P)$.