Higher integrality conditions, volumes and Ehrhart polynomials

TL;DR

This paper introduces the concept of k-integral polytopes, generalizing integral polytopes, and explores how their Ehrhart polynomial coefficients relate to projections and slices, revealing new volume relationships.

Contribution

It generalizes previous results by defining k-integral polytopes and linking Ehrhart coefficients to projections and slices, expanding understanding of polytope volume properties.

Findings

Coefficients in degrees ≤ k are determined by projections of the polytope.

Coefficients in degrees > k are determined by slices of the polytope.

Volume of a polytope equals the sum of volumes of its slices under certain conditions.

Abstract

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of $k$-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a $k$-integral polytope $P$ has the properties that the coefficients in degrees less than or equal to $k$ are determined by a projection of $P$, and the coefficients in higher degrees are determined by slices of $P$. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.