number is exactly the BKK bound of the system.

An important fact is that for generic choice of the coefficients, the BKK bound is exact.

\[lem:init-sys\] Let $F = (f_1,\dots,f_n)$ be a system of $n$ Laurent polynomials in $n$ variables. For generic choices of coefficients, and any nonzero $\mathbf{v} \in {\mathbb{R}}^n$ $\operatorname{init}_{\mathbf{v}} F$ has no solution in ${({\mathbb{C}}^*)}^n$.

\[rmk:generic\] “Generic choice” is a subtle concept in algebraic geometry. In the current context, it is sufficient to take the following “probability one” interpretation: If the coefficients are chosen at random (with independent distribution) among all possible complex coefficients, then with probability one, Lemma \[lem:init-sys\] is true. However, using the set of coefficients to parametrize a nonlinear system is not completely precise: For any nonsingular square matrix $A$, the system $F$ and $A \cdot F$ are naturally equivalent. Consequently, the more precise parametrization using a certain Grassmannian has to be considered in order to make sense of the concept of generic choice.

A relaxation of the BKK bound was developed in the context of algebraic synchronization equations [@chen_unmixing_2017] as well as the closely related “power-flow equations.” [@chen_network_2015].

\[def:adj-polytope\] Given a graph $G$, we define its **adjacency polytope** to be $$\nabla_G =
        \operatorname{conv}(\nabla_{G,1} 