I have originally posted [this question](https://math.stackexchange.com/questions/2447282/holomorphic-sards-theorem) 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?



Holomorphic Sard's theorem?
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)](https://mathoverflow.net/questions/2944), 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.


