Local $h^*$-polynomials for one-row Hermite normal form simplices

TL;DR

This paper studies the distribution of local $h^*$-polynomials for a specific class of lattice simplices in Hermite normal form, revealing a limiting distribution as volume increases.

Contribution

It establishes the existence of a limiting distribution for the coefficients of local $h^*$-polynomials in these simplices with fixed off-diagonal entries.

Findings

Distribution of coefficients converges as volume increases

Limiting distribution depends on a specific normalized volume

Analysis of two families illustrates main results

Abstract

The local $h^*$-polynomial of a lattice polytope is an important invariant arising in Ehrhart theory. Our focus is on lattice simplices presented in Hermite normal form with a single non-trivial row. We prove that when the off-diagonal entries are fixed, the distribution of coefficients for the local $h^*$-polynomial of these simplices has a limit as the normalized volume goes to infinity. Further, this limiting distribution is determined by the coefficients for a particular choice of normalized volume. We also provide an analysis of two specific families of such simplices to illustrate and motivate our main result.