Definition:
Let $\mathbb{A}=(A_1,\ldots,A_n)$ be the supports of the polynomials in $f=(f_1,\ldots,f_k)$ . Any nonzero vector $w$ defines the face system

$$\partial_w f =(\partial_w f_1,\ldots,\partial_w f_k)$$   with$$\quad \partial_w f_i=\sum_{a\in \partial_w A_i}c_a x^a$$

whose supports $\partial_w\mathbb{A}=(\partial_w A_1,\ldots,\partial_w A_n)$ span faces of the corresponding Newton polytopes. I.e.

$$\partial_w A_i := \{ a \in A_i\: :\: \langle a,w \rangle$$   min$$_{a'\in A_i}\langle a',w \rangle\} .$$

Theroem: [Bernshtein's second theorem]
If $\forall w\neq 0$ , $\partial_w f =0$ has no solution in $(\mathbb{C}^*)^n$ , then the mixed volume of the Newton polytopes of the $f_i$ gives the exact number of common zeros in $(\mathbb{C}^*)^n$ and all solutions are isolated. Otherwise it is a strict upper bound.