Zeros set of analytic functions over complex plane with several variables

Solution 1:

Identity principle: If two analytic functions $f,g$ on an open connected set $D\subseteq \mathbb{C}^n$ coincide on a nonempty open set $U\subseteq D$, then $f=g$ on $D$.

Proof: see here, Theorem 6, p. 6.

In particular, the zero set of a nonconstant analytic function on an open connected set has empty interior. In the case $n=1$, we have the stronger fact that the zeros of such functions are isolated. But this is no longer true for $n\geq 2$, as shown by the example $f(z_1,\ldots,z_n)=z_1$.

We actually have more than empty interior. It follows from Jensen's inequality that the zero set of a nonconstant analytic function on an open connected set has $2n$-dimensional Lebesgue measure zero. Same place, Corollary 10, p.9.