Let's for simplicity work in $\mathbb{C}^n$. Suppose that $f_1,\dots, f_n$ are polynomials and $0$ is an isolated solution of the system $f_1(z)=\dots=f_n(z)=0$. I want to bound from below the multiplicity of the intersection of corresponding divisors at the origin in terms of Newton polyhedra. For example, if all partial derivatives $\partial^a f_j$ vanishes at $0$ for every multi-index $a$ such, that $a_1/w_1+\cdots+a_n/w_n<m_j$, then the multiplicity of $0$ in the intersection is at least $m_1\cdots m_n w_1\cdots w_n$. This estimate essentially follows from Bernstein-Kushnirenko theorem. The derivation is not difficult, but it's probably well known. I am looking for references for this result.

In general case I guess, the answer should be in terms of mixed volumes of non-convex polyhedra (which are difference of two convex polhedra).