We show how to compute the Ehrhart polynomial of the free sum of two latticepolytopes containing the origin $P$ and $Q$ in terms of the enumerativecombinatorics 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 thatgiven a lattice polytope $P$ containing the origin, the problem of computingthe number of lattice points in all rational dilates of $P$ is equivalent tothe problem of computing the number of lattice points in all integer dilates ofall free sums of $P$ with itself.
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机译：我们展示了如何根据枚举组合$ P $和$ Q $计算包含原点$ P $和$ Q $的两个点阵多面体的自由总和的Ehrhart多项式。这概括了Beck，Jayawant，McAllister和Braun的工作，并从以下观察得出：加权的$ h ^ * $多项式相对于免费总和是可乘的。我们推论给定一个包含原点的格多边形$ P $，计算$ P $的所有有理扩张图中的格点数量的问题就等于计算$ P所有自由和的所有整数扩张中的晶格数量的问题。 $本身。
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