Local Formulas for Ehrhart Coefficients from Lattice Tiles
Maren H. Ring, Achill Schürmann
TL;DR
This work develops an infinite family of local formulas for Ehrhart coefficients by constructing region-based tilings in a lattice setting. Central to the approach is the domain complex $\text{DC}(Q)$ and a recursive region construction $R(C)$ tied to pointed rational cones, allowing the local weight function $\mu$ to be defined solely from normal cones $N_f$ and associated DC-volumes $v_C$ with correction terms $w^C_K$. By tiling the covering domain complex $\text{CDC}(t\mathcal{P})$ with translates of these regions and tracking feasible lattice points $\mathcal{X}(tf)$, the paper derives $|\Lambda\cap t\mathcal{P}|=\sum_f \mu(N_f)\text{vol}(tf)$ for large $t$, yielding a local formula for Ehrhart coefficients that can produce irrational values and can adapt to given polyhedral symmetries. The construction bypasses triangulations of cones and relies on basic polyhedral geometry, offering geometric interpretation and computational flexibility, including symmetry exploitation via Dirichlet–Voronoi cells.
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.