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Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices

Jhon B. Caicedo, Martina Juhnke, Germain Poullot

TL;DR

The paper advances the study of Ehrhart theory for ${\boldsymbol{s}}$-lecture hall simplices by identifying a new family ${\boldsymbol{s}}=(a,\dots,a,a+1)$ that becomes not Ehrhart positive as $a$ grows (for $n\ge5$), and by developing explicit, constructive frameworks for unimodular triangulations. It introduces a unifying one-point extension approach to extend known unimodular triangulation results and proves explicit cases where flag, regular, unimodular triangulations exist for both the (a,...,a,a+1) sequence and certain block-structured sequences with inserted ones. The authors also establish a decisive asymptotic negative coefficient result, conjecture that all ${\boldsymbol{s}}$-lecture hall polytopes admit flag, regular, unimodular triangulations, and provide algorithmsic, explicit triangulations, fostering deeper connections between Ehrhart theory, ${\boldsymbol{s}}$-Eulerian polynomials, and polyhedral combinatorics.

Abstract

Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit.

Ehrhart non-positivity and unimodular triangulations for classes of s-lecture hall simplices

TL;DR

The paper advances the study of Ehrhart theory for -lecture hall simplices by identifying a new family that becomes not Ehrhart positive as grows (for ), and by developing explicit, constructive frameworks for unimodular triangulations. It introduces a unifying one-point extension approach to extend known unimodular triangulation results and proves explicit cases where flag, regular, unimodular triangulations exist for both the (a,...,a,a+1) sequence and certain block-structured sequences with inserted ones. The authors also establish a decisive asymptotic negative coefficient result, conjecture that all -lecture hall polytopes admit flag, regular, unimodular triangulations, and provide algorithmsic, explicit triangulations, fostering deeper connections between Ehrhart theory, -Eulerian polynomials, and polyhedral combinatorics.

Abstract

Counting lattice points and triangulating polytopes is a prominent subject in discrete geometry, yet proving Ehrhart positivity or existence of unimodular triangulations remain of utmost difficulty in general, even for ``easy'' simplices. We study these questions for classes of s-lecture hall simplices. Inspired by a question of Olsen, we present a new natural class of sequences s for which the s-lecture hall simplices are not Ehrhart positive, by explicitly estimating a negative coefficient. Meanwhile, motivated by a conjecture of Hibi, Olsen and Tsuchiya, we extend the previously known classes of sequences s for which the s-lecture hall simplex admits a flag, regular and unimodular triangulation. The triangulations we construct are explicit.

Paper Structure

This paper contains 14 sections, 19 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Ehrhart (non) positivity
  3. Unimodular (flag, regular) triangulations
  4. Preliminaries
  5. Ehrhart polynomial and $h^\ast$-vector
  6. Triangulations
  7. ${\boldsymbol{s}}$-lecture hall simplices and ${\boldsymbol{s}}$-Eulerian polynomials
  8. Triangulating one-point extensions
  9. Ehrhart non-positivity
  10. Regular, flag and unimodular Triangulation
  11. Numerical values
  12. $\lambda_{a, b}$ for the proof of \ref{['eq:Eulerian_property']}
  13. Explicit values of $\beta(n)$ as defined in \ref{['ex:negative']}
  14. Triangulation of ${\mathcal{P}}_3^{(3,5,2)}$ from \ref{['example:wherever']}

Key Result

Theorem A

For any $n\geq 5$, and any $a$ big enough, the simplex $\mathcal{P}_n^{{\boldsymbol{s}}}$ is not Ehrhart positive for the sequence ${\boldsymbol{s}} = (a,\dots, a, a + 1)$ of length $n$. More precisely, as $a\to \infty$, we have: where $[t^{n-4}]\mathcal{L}_{\mathcal{P}_n^{{\boldsymbol{s}}}}(t)$ denotes the coefficient of $\mathcal{L}_{\mathcal{P}_n^{{\boldsymbol{s}}}}(t)$ in front of $t^{n-4}$.

Figures (3)

  • Figure 1: (Left & Middle) Two different unimodular triangulations of ${\mathcal{P}}_3^{{\boldsymbol{s}}}$ without its vertex ${\boldsymbol{v}}_3 = (1, 2, 3)$. (Right) A non-unimodular triangulation for ${\mathcal{P}}_3^{{\boldsymbol{s}}}$ without its vertex ${\boldsymbol{v}}_3 = (1, 2, 3)$.
  • Figure 2: Triangulation of $\mathcal{P}_n^{{\boldsymbol{s}}'}$ with ${\boldsymbol{s}}' = (s_1, \ldots, s_n + 1)$.
  • Figure 3: (Left) For ${\boldsymbol{s}} = (3,\, 2\cdot 3+1)$, only $\textcolor{blue!50}{\mathsf{R}_2}$ appears, with a single point added. (Right) The subpolytopes $\textcolor{blue!50}{\mathsf{R}_2}$ and $\textcolor{violet!50}{\mathsf{R}_3}$ for ${\boldsymbol{s}} = (3,\, 2\cdot 3+2)$.

Theorems & Definitions (53)

  • Theorem A: \ref{['The:Negativity']}
  • Conjecture A: hibi2016
  • Theorem B: \ref{['cor:extended', 'coro:join', 'Remark:dilation']}
  • Conjecture B
  • Lemma 2.1: beck2015computing
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 43 more