Asymptotics of Ehrhart Δ - Vectors

Let N be a lattice and P ⊂ N ⊗ Z R a lattice polytope, i.e., the convex hull of finitely many points in N. Ehrhart's theorem asserts that the lattice-point counting function f P (m) := # (mP ∩ N) is a polynomial, and thus 1 + P m≥1 f P (m) t m = δ P (t) (1−t) d+1 for some polynomial δ P (t), the δ-vector of P. Motivated by the Knudsen–Mumford–Waterman Conjecture about the existence of unimodular triangulations for all sufficiently large dilates of P and a recent theorem of Athanasiadis–Hibi–Stanley asserting that such triangulations imply certain inequalities for the δ-vector, we study δ nP (t) = δ 0 (n) + δ 1 (n) t + · · · + δ d (n) t d as n grows. We prove that for sufficiently large n (with a bound depending only on dim P), δ 0 (n) < δ d (n) < δ 1 (n) < · · · < δ i (n) < δ d−i (n) < δ i+1 (n) < · · · < δ ⌊ d+1 2 ⌋ (n); in particular, δ nP (t) is unimodal.