An upper estimate can be obtained from Proposition 1.1 in the following paper:

Rusek, Kamil; Winiarski, Tadeusz
Polynomial automorphisms of $\mathbb{C}^n$.
Univ. Iagel. Acta Math. No. 24 (1984), 143–149

http://www2.im.uj.edu.pl/actamath/PDF/24-143-149.pdf

Let $F=(F_1,...,F_n):\mathbb{C}^n \mapsto \mathbb{C}^n$, where $F_1,...,F_n$ are polynomials  (none of them identically zero). Assume that  $F^{-1}(0)=\{a_1,...a_k\}$. Then $\nu_F:= \sum_i m_{a_i}F \leq {\rm deg }F_1\cdot ...\cdot {\rm deg }F_n$, where $m_{a_i}F$ is the multiplicity of $F$ at the point $a_i$.

Note that the polynomials in the proposition are not necessarily homogeneous. The assumption that $F^{-1}(0)$ be finite ensures that the multiplicities are well defined.