# Zeros set of analytic functions over complex plane with several variables

I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane with several variables.

My question is: How I can understood this structure.

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Your first statement is wrong, take the constant zero function. for the second point I guess the trivial extension is that the zeroes are closed sets which only open subset is the empty set –  Dominic Michaelis Apr 5 '13 at 14:07

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 $D$.
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.