Given integers $d_1,\ldots,d_n$, is there an elementary way to justify that for generic polynomials $f_1,\ldots,f_n\in\mathbb{C}[X_1,\ldots,X_n]$ of degree less or equal than $d_1,\ldots,d_n$, the number of solutions of $f_1=\cdots=f_n=0$ is bounded by some constant $C(d_1,\ldots,d_n)$?
Explicit bound is given by Berstein's theorem for example, but I am interested by more basic argument.