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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…

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    $\begingroup$ Singularities of analytic functions in dimension $n\geq 2$ cannot be isolated. Two functions $f,g$ are called conjugate if $\phi\circ f\circ\phi^{-1}=g$. Please edit your question. $\endgroup$
    – Alexandre Eremenko
    Sep 17, 2013 at 13:09
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    $\begingroup$ @AlexandreEremenko: I'm not referring to dynamical conjugacy. I gave the meaning of conjugacy I'm after, that is $g=f\circ\phi$. This setting has been studied by 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 $f(x,y)=xy$ is isolated at $(0,0)$. That being said, you might want to reconsider your downvote ;) $\endgroup$
    – Loïc Teyssier
    Sep 17, 2013 at 13:24
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    $\begingroup$ Teyssier: I recondisedered my downvote;-) though your usage of the term "conjugate" is very unusual. $\endgroup$
    – Alexandre Eremenko
    Sep 17, 2013 at 22:27
  • $\begingroup$ Well, it's the conjugation for the right-action of the group of local change of coordinates, so from an algebraic point of view it's standard. Yet I agree that in general, analysis people (as I am) tend to think to the dynamical conjugation. $\endgroup$
    – Loïc Teyssier
    Sep 18, 2013 at 6:05
  • $\begingroup$ In the book by Greuel, Lossen and Shustin Introduction to singularities and deformations (page 118) this is called right 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, 2013 at 7:38

1 Answer 1

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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.

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