Counting lattice points in free sums of polytopes

We show how to compute the Ehrhart polynomial of the free sum of two lattice polytopes containing the origin P and Q in terms of the enumerative combinatorics of P and Q. This generalizes work of Beck, Jayawant, McAllister, and Braun, and follows from the observation that the weighted h⁎-polynomial is multiplicative with respect to the free sum. We deduce that given a lattice polytope P containing the origin, the problem of computing the number of lattice points in all rational dilates of P is equivalent to the problem of computing the number of lattice points in all integer dilates of all free sums of P with itself.