The Ehrhart Function for Symbols

Then Ehrhart’s theorem asserts that p(N, f) is a polynomial in N . In the case that ∆ is a simple polytope (meaning that n edges emanate from each vertex) Ehrhart’s theorem is a consequence of the Euler-MacLaurin formula, [Kh, KP, CS1, CS2, Gu, BV, DR] and one can be more explicit about the nature of the polynomial p(N, f). Let us explain how this works in the more restrictive case where ∆ is not only simple but is regular, meaning that the local cone at each vertex can be transformed by an integral unimodular affine transformation into a neighborhood of the origin in the standard orthant R+. 1 In this case we can apply the formula of KhovanskiiPukhlikov [KP], which reads as follows: The polytope ∆ can be described by a set