Bennet Goeckner was partially supported by an AMS-Simons travel grant. Martha Yip was partially supported by a Simons Collaboration grant.

Background {#Section:Background}
==========

In this section, we briefly recall notions from convex geometry, Ehrhart theory, and the study of Newton polytopes in algebraic combinatorics.

Convex polytopes and Ehrhart theory
-----------------------------------

A [**polytope**]{} ${\mathcal{P}}\subset {\mathbb{R}}^m$ is the convex hull of finitely many points $\bv_1,\ldots,\bv_k\in{\mathbb{R}}^m$. That is, $${\mathcal{P}}= \operatorname{conv}\{\bv_1,\ldots,\bv_k\}\coloneqq 
    \left\{ \x=\sum_{i=1}^k \nu_i \bv_i \ \bigg| \ 0\leq \nu_i\leq 1 \mbox{ and } \sum_{i=1}^k\nu_i=1\right\}.$$ The inclusion-minimal set $V\subseteq {\mathbb{R}}^m$ such that ${\mathcal{P}}=\operatorname{conv}(V)$ is called the [**vertex set**]{} of ${\mathcal{P}}$. A polytope is called [**lattice**]{} (resp. [**rational**]{}) if ${\mathcal{P}}=\operatorname{conv}(V)$ for $V\subseteq {\mathbb{Z}}^m$ (resp. $V\subseteq {\mathbb{Q}}^m$). Given a polytope ${\mathcal{P}}\subseteq{\mathbb{R}}^m$, the classical Minkowski–Weyl theorem states that we can express ${\mathcal{P}}$ as a bounded set of the form $${\mathcal{P}}=\{\x\in{\mathbb{R}}^m \mid \langle \a_i,\x \rangle \leq b_i \mbox{ for } i=1,\ldots,\ell\}$$ where $\langle \a_i,\x \rangle = \sum_{j=1}^m a_{ij}x_j$ for some $\a_1,\ldots,\a_\ell\in{\mathbb{R}}^m$ and $b_1,\ldots,b_\ell\in {\mathbb{R}}$. If none of these inequalities are redundant, these inequalities define the [**facets**]{}, or codimension 1 faces of ${\mathcal{P}}$. The [**dimension**]{} of ${\mathcal{P}}$, denoted $\dim({\mathcal{P}})$, is defined to be the dimension of its affine span in ${\mathbb{R}}^m$.

Let ${\mathcal{P}}$ be a lattice polytope with $\dim({\mathcal{P}})=d\leq 