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Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$.

If $n=1$, the zeros set consists of isolated points. How to generalize this to $n$ dimensions?

I know that $Z(f)$ has zero Lebesgue measure in $2n$ dimensional space. But this does not help: for $n=1$ it includes both isolated points and line segments.

Can the concept of limit/isolated/accumulation point be usefully generalized in this context? What's the best way to characterize $Z(f)$?

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Informally, the zero set of a holomorphic function that does not vanish identically (let's suppose it is defined on a domain, i.e. a connected open set) is a locally finite union of complex $(n-1)$-dimensional surfaces; after a change of variables, these are the graphs of holomorphic functions of $n-1$ variables. The simple example $f(z_1,z_2) = z_1\cdot z_2$ shows that it is in general not a submanifold, but most points of the zero set have a neighbourhood $U$ (in $\mathbb{C}^n$) such that $U\cap Z(f)$ is a submanifold of $U$, the set of non-regular points has lower dimension (than $n-1$).

The zero set of a non-constant holomorphic function does not locally disconnect $\mathbb{C}^n$: If $D\subset \mathbb{C}^n$ is a connected open subset, and $f \in \mathcal{O}(D)$, then $D\setminus Z(f)$ is connected [trivially, if $f\equiv 0$].

Zero sets of holomorphic functions of several complex variables are usually investigated (in more or less depth) in textbooks on several complex variables, e.g. R.M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer Verlag, GTM 108, Chapter I, Section 3, or in Grauert/Fritzsche, Einführung in die Funktionentheorie mehrerer Veränderlicher (English translation: Several Complex Variables), Springer Verlag, Chapter III, Section 6, there is a more extensive treatment of zero sets.

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  • $\begingroup$ Okey, good, thanks. $\endgroup$ – eli Jan 6 '14 at 1:36

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