TITLE: Local polynomial form of holomorphic functions
QUESTION [8 upvotes]: 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…

REPLY [11 votes]: 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.