I have a rational map $f:\mathbb C^n\longrightarrow \mathbb C^n,$ all I know $f$ is defined by homogenious polynomials of degree $m$ and $f$ not necessarily a morphism. Computer packages aside, I am wondering if the passonate algebraic geometers have a general scheme of computing $\deg f$ explicitly?

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/24143149.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. 

