Analytic order of singular and critical points

Shustin, Eugenii

doi:10.1090/S0002-9947-03-03409-3

Analytic order of singular and critical points
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by Eugenii Shustin

Trans. Amer. Math. Soc. 356 (2004), 953-985

DOI: https://doi.org/10.1090/S0002-9947-03-03409-3

Published electronically: August 21, 2003

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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 $\sqrt {\mu }+1\le d\le \mu +1$, where $\mu$ is the Milnor number. Our result in both the problems is $d\le a\sqrt {\mu }$ 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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