$S \subset {\mathbb{Z}}^n$. The set $\operatorname{conv}(S) \subset {\mathbb{R}}^n$ is called the *Newton polytope* of $f$. Given a nonzero $\mathbf{v} \in {\mathbb{R}}^n$, $\operatorname{init}_{\mathbf{v}} f$ is defined to be $\sum_{\mathbf{a} \in {(S)}_{\mathbf{v}}} c_{\mathbf{a}} \mathbf{x}^{\mathbf{a}}$ where ${(S)}_{\mathbf{v}}$ is the subset of $S$ on which the linear functional $\langle \cdot, \mathbf{v} \rangle$ attains its minimum. Extending this notation to a system of Laurent polynomials $F = (f_1,\dots,f_n)$, we write $\operatorname{init}_{\mathbf{v}} F = (\operatorname{init}_{\mathbf{v}} f_1,\dots,\operatorname{init}_{\mathbf{v}} f_n$). Newton polytopes play critical roles in calculating the generic number of isolated solutions in ${({\mathbb{C}}^*)}^n$ (or simply ${\mathbb{C}}^*$-solutions) a system of $n$ Laurent polynomial equations could have. Indeed, this generic ${\mathbb{C}}^*$-solution count is given by the mixed volume of the Newton polytopes. This is the content of Bernshtein’s Theorem [@bernshtein_number_1975; @khovanskii_newton_1978; @kushnirenko_newton_1976], and this count has since been known as the **BKK bound** [@canny_optimal_1991]. Though we will not directly compute BKK bounds, the condition for BKK bound to be exact will be used in establishing our main results.

\[thm:bernshtein-b\] Consider a system of $n$ Laurent polynomials $F = (f_1,\dots,f_n)$ in $n$ variables. If $\operatorname{init}_{\mathbf{v}} F$ has no solution in ${({\mathbb{C}}^*)}^n$ for any nonzero vector $\mathbf{v} \in {\mathbb{R}}^n$, then all solutions of $F(\mathbf{x}) = \mathbf{0}$ are isolated and the total 