An Ehrhart Series Formula For Reflexive Polytopes

It is well known that for P and Q lattice polytopes, the Ehrhart polynomial of P×Q satisfies LP×Q(t) = LP (t)LQ(t). We show that there is a similar multiplicative relationship between the Ehrhart series for P , for Q, and for the free sum P ⊕ Q that holds when P is reflexive and Q contains 0 in its interior. Let P be a lattice polytope of dimension d, i.e. a convex polytope in R whose vertices are elements of Z and whose affine span has dimension d. A remarkable theorem due to E. Ehrhart, [4], asserts that for non-negative integers t the number of lattice points in the t dilate of P is given by a degree d polynomial in t denoted by LP (t) and called the Ehrhart polynomial of P . We let EhrP (x) = ∑ t≥0 LP (t)x t = ∑d j=0 h ∗ jx j (1 − x)d+1 denote the rational generating function for this polynomial (as in [7], chapter 4), called the Ehrhart series of P . See [3] for more information regarding Ehrhart theory. In this note we are concerned with a multiplicative decomposition for EhrF (x) when F is a free sum of two lattice polytopes subject to some restrictions on the summands. For two polytopes P ⊆ RP and Q ⊆ RQ of dimension dP and dQ, define the free sum to be P ⊕ Q = conv{(0P × Q) ∪ (P × 0Q)} ⊆ R dP +dQ . Let P = { x ∈ RP : x · p ≤ 1 for all p ∈ P } denote the dual of P and P ◦ denote the interior of P . If 0 ∈ P ◦ and 0 ∈ Q, then the free sum operation is dual to the product operation, i.e. (P ×Q) = (P)⊕(Q), [5]. A basic example of this duality can be seen with the dual polytopes given by the d-dimensional crosspolytope and the d-dimensional cube, being the free sum of d copies of the interval [−1, 1] and the product of d copies of [−1, 1], respectively. the electronic journal of combinatorics 13 (2006), #N15 1 From the perspective of Ehrhart theory, it is natural to ask which lattice polytopes have duals that are also lattice polytopes and hence have Ehrhart polynomials. A polytope with this property is called reflexive. Free sums of reflexive polytopes have recently played a crucial role in [2]. Reflexive polytopes have many rich properties, as seen in the following lemma. Lemma 1 ([1] and [6]) P is reflexive if and only if P is a lattice polytope with 0 ∈ P ◦ that satisfies one of the following (equivalent) conditions: i. P is a lattice polytope. ii. LP ◦(t + 1) = LP (t) for all t ∈ N, i.e. all lattice points in R dP sit on the boundary of some non-negative integral dilate of P . iii. hi = h ∗ dP −i for all i, where hi is the i th coefficient in the numerator of the Ehrhart series for P . For a product of polytopes, it is easy to see that LP×Q(t) = LP (t)LQ(t). The following theorem indicates that Ehrhart polynomials also behave nicely for the free sum if a reflexive polytope is involved. Theorem 1 If P is a dP -dimensional reflexive polytope in R dP and Q is a dQ-dimensional lattice polytope in RQ with 0 ∈ Q, then EhrP⊕Q(x) = (1 − x)EhrP (x)EhrQ(x). (1) The key point in the following proof is that the RP and RQ components of lattice points in t(P⊕Q) cannot simultaneously be far from the origin. For what follows, consider vectors in P and Q as actually being in P ⊕ 0Q and 0P ⊕ Q, respectively. Proof: Note that (1) is equivalent to LP⊕Q(t) = LQ(t) + t ∑ k=1 LQ(t − k)(LP (k) − LP (k − 1)) (2) for every t ∈ N. This equivalence is seen by expanding the product on the right hand side of (1) as follows: (1 − x)EhrP (x)EhrQ(x)xi = (1 − x)( ∑