Arithmetic aspects of symmetric edge polytopes

TL;DR

This paper explores the arithmetic, geometric, and combinatorial properties of symmetric edge polytopes, providing explicit formulas and proving conjectures related to their $h^*$-polynomials, especially for complete bipartite graphs.

Contribution

It offers a complete combinatorial description of facets and an explicit formula for the $h^*$-polynomial of symmetric edge polytopes of complete bipartite graphs, proving conjectures on their properties.

Findings

$h^*$-polynomial is $ ext{γ}$-positive and real-rooted

Proves Gal's conjecture for flag unimodular triangulations

Strengthens results by Nevo and Petersen (2011)

Abstract

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gr\"obner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^\ast$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^\ast$-polynomial is $\gamma$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).