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arXiv · 2608.03635

Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials

Benjamin Nill
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Abstract

In this note we prove that in fixed dimension the Ehrhart $h^*$-polynomial of a lattice polytope of sufficiently large lattice width is real-rooted. In particular, this implies strict log-concavity and unimodality of the $h^*$-vector and answers a question of Averkov, Hofscheier and the author. For a lattice simplex we prove the analogous statement for its local $h^*$-polynomial, also called box polynomial. The proofs were found using ChatGPT 5.6 Sol and follow essentially directly from a result by Basu and Oertel that for large enough lattice width counting lattice points approximates the volume.

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Benjamin Nill. 2026-08-04. Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials. https://arxiv.org/abs/2608.03635

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