q-Polynomials
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
Rd), we fix an integral linear form l and sum ql(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.
Our plan is to start with an introduction to Ehrhart theory, including
recent results. Second, 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. Third, we will 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.
Novel results in these lectures are based on joint projects with Esme Bajo,
Ben Braun, Alvaro Cornejo, Thomas Kunze, and Andrés Vindas-Melėndez.