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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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    $\begingroup$ 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 $\endgroup$ – Dominic Michaelis Apr 5 '13 at 14:07
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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.

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    $\begingroup$ Thank you very much for your precise answer to E. James's question, and above all for having let me know the wonderful book "Analytic Functions of Several Variables" by Gunning and Rossi. $\endgroup$ – Maurizio Barbato Jan 13 '17 at 19:28

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