| If P is a lattice polytope (i.e., P is the convex hull of finitely many integer points in R^d), Ehrhart's theorem asserts that the integer-point counting function L_P(m) = #(mP \cap Z^d) is a polynomial in the integer variable m. Equivalently, the generating function Ehr_P(t) = \sum_{m \ge 0} L_P(m) t^m is a rational function of the form h(t)/(1-t)^{d+1}; we call h(t) the Ehrhart h-vector of P. We study the behavior of the Ehrhart series Ehr_{nP}(t) = \sum_{m \ge 0} L_P(nm) t^m as n grows; e.g., we can prove that the Ehrhart h-vector of nP is eventually unimodal, where "eventually" only depends on the dimension of P. Our results are general combinatorial theorems about generating functions and can be applied to other settings, e.g., Veronese subrings of graded rings. This is joint work with Alan Stapledon (Michigan). |