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Let $f: \mathbb C ^2\to \mathbb C$ be (a germ at $0$ of) a holomorphic function. Does there exist a small neighborhood $U_0\in \mathbb{C}^2$ of $0$ and a holomorphic change of coordinates $g$ (i.e. an invertible holomorphic function $g:U_0 \to U$ to a neighborhood of 0 in $\mathbb C^2$), such that $fg^{-1}$ is a polynomial function?

The answer is yes, if the Weierstrass polynomial of $f$

$$f(z,w)=z^k(w^m+a_{m-1}(z)w^{m-1}+\dots+a_0(z))\cdot\text{unit}$$

has the non identically vanishing discriminant (as a polynomial in $w$ over holomorphic functions). This is a theorem of N. Levinson.

Is there an example of a germ of a holomorphic function (necessarily with quadratic factors in the Weierstrass polynomial) that is not equivalent to a germ of a polynomial in the above sense?

In particular, that would give an example of an (non-reduced) analytic divisor in a neighborhood of $0$ in $\mathbb{C^2}$ that cannot be made algebraic by a local holomorphic change of coordinates. For reduced analytic divisor it is always possible (see e.g. Bogomolov, Cascini, de Oliveira for a proof independent of the Levinson theorem)

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