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

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 $h^\ast(P)$.