Ehrhart-Macdonald reciprocity with multiplicities

Let $$P$$ be a convex lattice polytope in $$\mathbb{R}^n$$. The function $$L(t, P) = |\mathbb{Z}^n \cap t\cdot P|$$ is a polynomial, and we have an equality $$L(-t, P) = (-1)^nL(t, P^{int}),$$ where $$P^{int} = P \backslash \partial P$$, $$t \in \mathbb{Z}$$. Therefore if for some integer $$n \in \mathbb{Z}_{\geq 0}$$ the polytope $$n\cdot P$$ has no interior lattice points we can conclude that $$L(t, P)$$ is divisible by $$t + n$$.

Question: Are there generalizations that allow us to estimate multiplicity of the root $$-n$$?

Motivation: There are families of marked poset polytopes $$P$$ coming from representation theory of Lie algebras that have a very simple expression for $$L(t, P)$$ of the form $$C\cdot \prod (t + n)^{a_n}$$, but no straightforward proof of these combinatorial results is known.

• It would be interesting if there were some way to get at the multiplicities, but I have never seen anything like that. As you say, when Ehrhart polynomials (or even more specifically, order polynomials) factor is quite mysterious (you might be interested in my survey arxiv.org/abs/2006.01568). Apr 26, 2021 at 23:19
• Also, a bit of caution: arxiv.org/abs/1609.00647 Apr 27, 2021 at 6:23
• @SamHopkins , i read the paper carefully. I would like to mention that all examples consider single variable. Note that for GZ and FFLV polytopes there can be one variable for each positive simple root, and the product formula still exists, and the collection of posets which satisfy this is also weird. Also, may i ask you where can i find the proof of the case $AP(n, d, l)$ ? May 17, 2021 at 16:41
• @RybinDmitry: See Stanley's "Enumerative Combinatorics, Vol. 2," Exercise 7.101. I'm not sure there is a published proof anywhere. May 17, 2021 at 16:51