Proof of Chapoton's conjecture on Newton polytopes of \(q\)-Ehrhart polynomials

Abstract

Recently, Chapoton found a \(q\)-analog of Ehrhart polynomials, which are polynomials in variable \(x\) whose coefficients are rational functions in \(q\). Chapoton conjectured the shape of the Newton polytope of the numerator of the \(q\)-Ehrhart polynomial associated to an order polytope. In this paper, we prove Chapoton's conjecture.