Ehrhart positivity of Tesler polytopes and Berline-Vergne's valuation

TL;DR

This paper investigates the positivity of Ehrhart polynomial coefficients of Tesler polytopes, employing Berline-Vergne's valuation to establish positivity for certain coefficients and generalizing results to all deformations.

Contribution

It introduces a new method to compute Berline-Vergne's function values directly from facet normals and proves positivity of specific Ehrhart coefficients for Tesler polytopes and their deformations.

Findings

Proved positivity of the 3rd and 4th Ehrhart coefficients for Tesler polytopes.

Developed a method to compute Berline-Vergne's function from facet normal vectors.

Generalized positivity results to all deformations of Tesler polytopes.

Abstract

For $\ba \in \R_{\geq 0}^{n}$, the Tesler polytope $\tes_{n}(\ba)$ is the set of upper triangular matrices with non-negative entries whose hook sum vector is $\ba$. Motivated by a conjecture of Morales', we study the questions of whether the coefficients of the Ehrhart polynomial of $\tes_n(1,1,\dots,1)$ are positive. We attack this problem by studying a certain function constructed by Berline-Vergne and its values on faces of a unimodularly equivalent copy of $\tes_n(1,1,\dots,1).$ We develop a method of obtaining the dot products appeared in formulas for computing Berline-Vergne's function directly from facet normal vectors. Using this method together with known formulas, we are able to show Berline-Vergne's function has positive values on codimension $2$ and $3$ faces of the polytopes we consider. As a consequence, we prove that the $3$rd and $4$th coefficients of the Ehrhart polynomial of $\tes_{n}(1,\dots,1)$ are positive. Using the Reduction Theorem by Castillo and the second author, we generalize the above result to all deformations of $\tes_{n}(1,\dots,1)$ including all the integral Tesler polytopes.