  For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a
$\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations
coming from some algebraic partial differential equations such that the
dimension of $F(P)$ is the number of irreducible factors of $P$. Moreover, the
knowledge of $F(P)$ gives a complete factorization of the polynomial $P$ by
taking gcd's. This generalizes previous results by Ruppert and Gao in the case
$n=2$.
