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Question: What is the correct notion of a product of integral (or rational) polytopes which induces a factorization of its Ehrhart (quasi-)polynomial into two primitive Ehrhart (quasi-)polynomials corresponding to its constituent polytopes, viz., $L_{P \times Q}(t) = L_{P}(t) L_{Q}(t)$?

(Motivation) Given two closed integral polytopes $P$ and $Q$ each with vertices at $\{ \mathbf{0} , b_{1} \mathbf{e}_{1}, \dots, b_{n} \mathbf{e}_{n} \}$ and $\{ \mathbf{0} , d_{1} \mathbf{e}_{1}, \dots, d_{m} \mathbf{e}_{m} \}$, respectively, where $n, b_{i}, m, d_{j} \in \mathbb{N}$, define the integral polytope $R$ with vertices at $\{ \mathbf{0}, b_{1} \mathbf{e}_{1}, \dots, b_{n} \mathbf{e}_{n}, d_{1} \mathbf{e}_{n+1}, \dots, d_{m} \mathbf{e}_{n+m} \}$.

The above construction cannot be the sought after product $P \times Q$. Suppose $P$ and $Q$ are defined by $b_{1} = b_{2} = d_{1} = d_{2} = 2$. It is easy to show that $L_{P}(1) = L_{Q}(1) = 6$. Define $R$ as above with vertices of $P$ and $Q$. It is true that $L_{R}(1) = 15 \neq 6^{2}$.

Question: What is $R$ in terms of $P$ and $Q$? Is it special in some way?


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2 Answers 2

up vote 4 down vote accepted

Isn't this obvious (or have I missed something?)

The appropriate product of rational polytopes $P\subseteq\mathbb R^m$ and $Q\subseteq\mathbb R^n$ is surely their Cartesian product $P\times Q\subseteq\mathbb R^{m+n}$. For a positive integer $t$, the integer points in $t(P\times Q)$ are those of the form $(a,b)$ where $a\in tP$ and $b\in tQ$, so there are $L_P(t)L_Q(t)$ of them.

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Thanks for the post! Two questions: What are the $n m$ vertices of the cartesian product $P \times Q$? Does this factorization continue to hold for rational polytopes and their quasi-polynomials (whose coefficient functions may have distinct periods)? –  user02138 Nov 23 '10 at 16:14
The vertices of $P\times Q$ are $(a,b)$ where $a$ is a vertex of $P$ and $b$ is a vertex of $Q$. In this answer I didn't assume that $P$ and $Q$ had integer point vertices. –  Robin Chapman Nov 23 '10 at 16:57

The (Motivation) description appears to be similar to a direct sum (a.k.a. tegum product) as defined in Section 1.2 of this paper:


The difference is that for a direct sum, 0 is in the relative interior of both of the factor polytopes, rather than a common vertex.

The same article answers your question about what the nm vertices of a Cartesian product are.

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