I've heard it stated that if you have a family $p_{1}, \ldots, p_{\ell} \in \mathbb{C}[x_1 , \ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = \bigcap_{j = 1}^{\ell} p_{j}^{-1} (\{0\}) \subseteq \mathbb{C}^n$ is either finite, or not compact in the standard topology on $\mathbb{C}^n$. My understanding is that this claim can be demonstrated by algebraic-geometric means, though the methods are rather involved. Can it be proven by analytic methods? I assume that it won't be a simple or easy argument, but can it be reasonably done?
Thanks!
EDIT: We've figured out how to show that if $p$ is nonconstant, then the null set of $p$ is either finite (if $n = 1$) or unbounded (if $n > 1$). But that's it.