if its dual $$P^* = \{ \mathbf{x} \in {\mathbb{R}}^n \mid \langle \mathbf{x}, \mathbf{p} \rangle \ge -1,\; \forall \mathbf{p} \in P \}$$ is also a lattice polytope. Given two convex polytopes $P \subset {\mathbb{R}}^n$ and $Q \subset {\mathbb{R}}^m$ both containing the origin, their *free sum*, denoted $P \oplus Q$, is $\operatorname{conv}( P' \cup Q' ) \subset {\mathbb{R}}^{n+m}$ where $$P' = \{ (\mathbf{p}, \mathbf{0}) \in {\mathbb{R}}^{n+m} \mid \mathbf{p} \in P \}$$ and $$Q' = \{ (\mathbf{0}, \mathbf{q}) \in {\mathbb{R}}^{n+m} \mid \mathbf{q} \in Q \}.$$ An important fact is that under mild conditions, the normalized volume of a free sum of lattice polytopes factors.

\[lem:free sum-vol\] Given two convex lattice polytopes $P$ and $Q$ both containing the origin as an interior point, if one of them is reflexive, then $\operatorname{NVol}(P \oplus Q) = \operatorname{NVol}(P) \cdot \operatorname{NVol}(Q)$.

The set ${({\mathbb{C}}^*)}^n$, known as an *algebraic torus*, has the structure of an abelian group under component-wise multiplication, and it will be the space in which we study the root count of synchronization equations. A *Laurent monomial* in $\mathbf{x}=(x_1,\dots,x_n)$ induced by vector $\mathbf{a} = (a_1,\dots,a_n) \in {\mathbb{Z}}^n$ is the formal expression $\mathbf{x}^{\mathbf{a}} = x_1^{a_1} \cdots x_n^{a_n}$. It is easy to verify that as a map from ${({\mathbb{C}}^*)}^n$ 