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arXiv · 1807.07678

Arithmetic aspects of symmetric edge polytopes

Akihiro Higashitani·Katharina Jochemko·Mateusz Michałek
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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).

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Akihiro Higashitani, Katharina Jochemko, Mateusz Michałek. 2018-07-20. Arithmetic aspects of symmetric edge polytopes. https://doi.org/10.1112/s0025579319000147

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