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, $t_i\in \mathbb Z$
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$\begingroup$ Google for "mixed volume" $\endgroup$– Thomas KahleCommented May 6, 2013 at 6:24
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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
answered May 6, 2013 at 14:24
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$\begingroup$ Thank you very much. I also have question about the first and the second coefficients of this multivariate polynomial, since we have known them in the case univariate. Could you please explain me a little bit about changing computation of the second coefficients in Ehrhart polynomial from boundary volume to normal vectors on $\mathbb S^n$. It is on the page 56, lemma 3.8 of this dissertation google.com.vn/… $\endgroup$ Commented May 8, 2013 at 9:44