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It is equivalent to show that the complement, say $V$, has empty interior. In fact $V$ has measure zero (say, with respect to Lebesgue measure on $\mathbb{C}^n$; any measure which is absolutely continuous with respect to Lebesgue measure would serve as well). If $V$ is smooth, this is very standard: the Implicit Function Theorem (or something like that). In general, you can use the fact that $V$ has a relatively open subset which is a smooth manifold whose complement has positive codimension and finish off by induction.