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