It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (*i.e.* the singularity of the analytic variety defined by the zero locus of the function), is locally conjugate (by right composition with a local biholomorphism) to a polynomial (in fact, a finite jet of the function).

This statement is not true anymore when $n\geq 3$ if the singularity is not isolated, although I can't find a reference containning examples of germs not locally conjugate to polynomials. Yet if I remember correctly, it is true when $n=2$.

Also I would be interested in knowing what happens in the meromorphic case, which I believe is still open.

I would be grateful for any pointer in the litterature towards such results!

PS: I hope this question is not a duplicate. I couldn't possibly browse through all the search results for requests like «polynomial holomorphic function», and was unable to think about sharper keywords…

e.g.Arnold, Mather, Tougeron, Yau. Besides I think you mistook the notion of singularity I'm referring to, that of tha analytic variety $\{f=0\}$, I edited the question to remove any ambiguity. In that setting the singularity of $f(x,y)=xy$isisolated at $(0,0)$. That being said, you might want to reconsider your downvote ;) $\endgroup$ – Loïc Teyssier Sep 17 '13 at 13:24Introduction to singularities and deformations(page 118) this is calledright equivalence. Actually, I've never heard of the term "conjugation" used in this context, but I must say that I'm not really an expert in the field. $\endgroup$ – Francesco Polizzi Sep 18 '13 at 7:381more comment