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…



Local polynomial form of holomorphic functions
The following example of non-algebraic germ is due to Whitney, see 


H. Whitney: *Local properties of analytic varieties*, Differential and Combinatorial Topology 205–244, Princeton University Press (1965). 


Take the germ of singularity in $\mathbb{C}^3$ given by $f(x,y,z)=0$, where $$f=xy(x+y)(x- zy)(x-e^zy).$$
The set $\{f=0\} \cap \{z= \lambda \}$ is given by five distinct lines through the origin.
If $f$ would be conjugate to a polynomial, the cross-ratios of any four of these lines would depend algebraically on $\lambda$. But this is impossible, because we have $\lambda$ and $e^{\lambda}$.


So this germ is *not* conjugate to a polynomial (or algebraic) germ. 


