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

Esme Bajo, Benjamin Braun, Giulia Codenotti, Johannes Hofscheier, Andrés R. Vindas-Meléndez

TL;DR

We address the problem of understanding local $h^*$-polynomials for one-row Hermite normal form simplices by fixing the off-diagonal entries and letting the normalized volume grow. The authors develop a parity-age framework via the parallelepiped group to define $B(S;z)$ and establish a central limit-type phenomenon: as $N\to\infty$ the coefficient distribution of $B(S;z)/B(S;1)$ converges to that at a finite volume $N=M+1$, where $M=\text{lcm}(a_1,\tcdots,a_{d-1},-1+\sum a_i)$. They illustrate the general theory with two families— the all-ones and geometric-sequence simplices—proving precise unimodality results and showing that unimodality can occur without the integer decomposition property. A key technical component is a suite of floor/ceiling lemmas together with Stapledon decompositions linking local and boundary $h^*$-polynomials. The work provides asymptotic behavior for a broad class of local invariants in Ehrhart theory and raises questions about typical unimodality and multi-row generalizations.

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.

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

TL;DR

We address the problem of understanding local -polynomials for one-row Hermite normal form simplices by fixing the off-diagonal entries and letting the normalized volume grow. The authors develop a parity-age framework via the parallelepiped group to define and establish a central limit-type phenomenon: as the coefficient distribution of converges to that at a finite volume , where . They illustrate the general theory with two families— the all-ones and geometric-sequence simplices—proving precise unimodality results and showing that unimodality can occur without the integer decomposition property. A key technical component is a suite of floor/ceiling lemmas together with Stapledon decompositions linking local and boundary -polynomials. The work provides asymptotic behavior for a broad class of local invariants in Ehrhart theory and raises questions about typical unimodality and multi-row generalizations.

Abstract

The local -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 -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.
Paper Structure (13 sections, 26 theorems, 125 equations, 6 figures, 1 table)

This paper contains 13 sections, 26 theorems, 125 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Lattice simplices and our contributions
  4. Properties of Local --Polynomials
  5. Hermite normal form simplices
  6. -- and local --polynomials
  7. Relationship to Stapledon decompositions
  8. Two Illustrative Examples
  9. Non-trivial row -
  10. Non-trivial row -
  11. Number Theoretical Results on Floor and Ceiling Functions
  12. Asymptotic Properties of Local --Polynomials
  13. Further Questions

Key Result

Theorem 1.1

Fix $a_1, \dots, a_{d-1} \in {\mathbb{Z}}_{\geq 1}$ and let $M \coloneqq \mathrm{lcm}(a_1, \dots, a_{d-1}, -1+\sum_{i=1}^{d-1}a_i)$. Let $S_N$ denote the one row Hermite normal form simplex for these values of $a_1, \dots, a_{d-1}$ and normalized volume $N$. As $N\to\infty$, the distributions for th

Figures (6)

  • Figure 1: The distribution of the coefficients of $B(S;z)/B(S;1)$ for the one-row Hermite normal form simplex $S$ with non-trivial row $(1,1,\ldots,1,331)$ in dimension $17$. Note that $331=22\cdot 15+1$ and $B(S;z)=22\cdot\sum_{i=2}^{16}z^i$.
  • Figure 2: The distribution of the coefficients of $B(S;z)/B(S;1)$ for the one-row Hermite normal form simplex $S$ with non-trivial row $(3^{11},3^{10},3^9,\ldots,3^2,3,1,3^{12})$ in dimension $13$.
  • Figure 3: The distributions for $100$ local $h^*$-polynomials of $11$-dimensional one-row Hermite normal form simplices with $N=505$. None of these simplices have the integer decomposition property.
  • Figure 4: The distribution of the values $\mleft\lceil{ak/N}\mright\rceil$ for $k=1,\dots,N$.
  • Figure 5: For each $n$, the fraction of unimodal local $h^*$-polynomials for one-row Hermite normal form simplices $S_{M+1}$ with one row given by a partition of $n$.
  • ...and 1 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Proposition 2.9
  • ...and 43 more