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Description

Chapoton initiated the study of q-Ehrhart polynomials: given a lattice polytope P (i.e., P is the convex hull of finitely many integer points in $\mathbb{R}^d$), we fix an integral linear form $l$ and sum $q^{l(m)}$ over all integer lattice points m in the dilate nP, as a function ehr(q,n). For q=1, this recovers the usual Ehrhart polynomial of P, counting integer lattice points in nP. From the viewpoint of the multigraded Hilbert series of the homogenization of P, the generating function of ehr(q,n) is a simple specialization. However, ehr(q,n) still carries an (a priori surprising) polynomial structure: Chapoton proved that that there exists a polynomial chap(x), whose coefficients are rational functions of q, such that ehr(q,n) equals the evaluation of chap(x) at the q-integer [n]_q.

We will show how Chapoton's results follow somewhat organically from Brion's Theorem, which decomposes the integer-point structure of P into that of its vertex cones.  This ansatz also yields several immediate extensions of Chapoton's work. We will also outline how similar q-polynomials might be useful in other settings, such as generalizing the chromatic polynomial of a graph, with connections to chromatic symmetric functions and the arithmetic of order cones.

This talk is based on joint work with Thomas Kunze.