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Analytic order of singular and critical points

Eugenii Shustin
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Abstract

We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a germ of a holomorphic function in two variables with an isolated critical point, what is the minimal degree of a polynomial, equivalent to the given function up to a local holomorphic coordinate change? Classically known estimates for such a degree $d$ in these questions are $\sqrtμ+1\le d\le μ+1$, where $μ$ is the Milnor number. Our result in both the problems is $d\le a\sqrtμ$ with an absolute constant $a$. As a corollary, we obtain asymptotically proper sufficient conditions for the existence of algebraic curves with prescribed singularities on smooth algebraic surfaces.

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Eugenii Shustin. 2002-09-04. Analytic order of singular and critical points. https://arxiv.org/abs/math/0209043

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