TITLE: Holomorphic Sard's theorem?
QUESTION [7 upvotes]: I have originally posted this question on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:U\to V$ be a holomorphic map.

Is it true that there is a non-zero holomorphic function $w$ on a neighbourhood of $\Phi(U)$, which vanishes at every critical value of $\Phi$?

If this is wrong, is there any other refinement of Sard's theorem for the holomorphic maps between complex domains?

REPLY [8 votes]: No.  We can enumerate $\mathbb{Q}[i]$ as $\{a_0,a_1,a_2,\dotsc\}$ and then choose a holomorphic function $f\colon\mathbb{C}\to\mathbb{C}$ with $f(n)=a_n$ and $f'(n)=0$ for all $n$.  This follows from a well-known interpolation theorem; some references are discussed at Which sequences can be extended to analytic functions? (e. g., Ackermann's function), for example.  Now the set of critical values is dense, so you probably cannot do much better than Sard's conclusion that it has measure zero.