2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157808For 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 $n=2$.Accepted in Mathematica Scandinavica. 8 pagesAlgebraic GeometryAlgebraic Topology12D05 (Primary) 14F40,14J70(Secondary)text