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.