We have known from Ehrhart theory that if $P$ is a $d$-dimensional polytope of $\mathbb R^n$ which has integer vertices then the number of integer points in $nP$ is a polynomial of degree $d$. We also know the leading coefficients, the second and the constant coefficients. I wonder if we have a similar conclusion in the case we count the number of integer points in $t_1P_1+...+t_kP_k$ where + denotes Minkowski sum and $P_i$ are polytopes of interger vertices, $t_i\in \mathbb Z$
Proving this is actually problem 3 on page 164 of Integer Points in Polyhedra by Alexander Barvinok - the number of integer points is a polynomial in $t_1,...,t_k$ as long as they are non-negative integers.
Proof omitted at the moment because I'm too rusty to produce one