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Let $f$ be a holomorphic function in $n$ complex variables on a domain $D\subset \mathbb C^n$. Let $S$ be a subset of $D$ such that for a polynomial $P$ in $n$ variable, $P(S)=(a,b)$ for an interval $(a,b)$. Can $f$ be zero on $S$?

What happens if $P(S)=(a,b)\cap \mathbb Q$?

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What does this subset look like? Can it be a Cantor-like set? Non-measurable? Semi-locally simply-connected? –  David Roberts Nov 3 '11 at 6:17
    
$D$ can be taken to be an infinite convex domain like a tube in $\mathbb C^n$. Take for instance $P(z)=z_1^2+\dots +z_n^2+A$ with $A\in \mathbb R$. Then $P$ will map $D$ in a parabolic region say $PARB$ on $\mathbb C$. Now $S$ is the pre-image of the set of points on $\mathbb R\cap PARB$. Can $S$ be as bad as a cantor set in this case? –  spr Nov 3 '11 at 6:41
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You need to specify what you allow for $S$. By just saying 'subset' you open yourself up to all sorts of pathologies. –  David Roberts Nov 3 '11 at 8:09
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I don't understand the question, maybe. $0$ is a holomorphic function, and it's going to be zero on any set $S$. So let's assume you mean "can $S$ be $f^{-1}(0)$?. Well, a polynomial (or a holomorphic function generally) cannot be real and nonconstant on this $n-1$ dimensional complex analytic variety, if $n>1$. –  Tom Goodwillie Nov 3 '11 at 13:52
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Could you edit your question to elaborate on how this problem arose for you? That might yield clues for making the problem both nontrivial and tractable. For example, do you have a reason to require $f$ to be nonconstant, or $S$ to have a convex hull with nonempty interior? –  S. Carnahan Nov 3 '11 at 17:30

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