Local Formulas for Ehrhart Coefficients from Lattice Tiles

TL;DR

This paper develops a broad class of local formulas for Ehrhart coefficients of lattice polytopes, based on polyhedral volume computations and fundamental domains, offering geometric insights and symmetry-adapted options.

Contribution

It introduces an infinite family of local formulas for Ehrhart coefficients that do not depend on triangulations, expanding the geometric and symmetric understanding of these coefficients.

Findings

Provides a geometric interpretation of Ehrhart coefficients.

Constructs local formulas based on fundamental domains in sublattices.

Allows formulas to be adapted to polyhedral symmetry groups.

Abstract

As shown by McMullen in 1983, the coefficients of the Ehrhart polynomial of a lattice polytope can be written as a weighted sum of facial volumes. The weights in such a local formula depend only on the outer normal cones of faces, but are far from being unique. In this paper, we develop an infinite class of such local formulas. These are based on choices of fundamental domains in sublattices and obtained by polyhedral volume computations. We hereby also give a kind of geometric interpretation for the Ehrhart coefficients. Since our construction gives us a great variety of possible local formulas, these can, for instance, be chosen to fit well with a given polyhedral symmetry group. In contrast to other constructions of local formulas, ours does not rely on triangulations of rational cones into simplicial or even unimodular ones.