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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analytic order of singular and critical points


Author: Eugenii Shustin
Journal: Trans. Amer. Math. Soc. 356 (2004), 953-985
MSC (2000): Primary 14F17, 14H20; Secondary 58K05
Published electronically: August 21, 2003
MathSciNet review: 1984463
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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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Additional Information

Eugenii Shustin
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
Email: shustin@post.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03409-3
PII: S 0002-9947(03)03409-3
Received by editor(s): July 5, 2002
Published electronically: August 21, 2003
Additional Notes: The author was partially supported by Grant No. G-616-15.6/99 of the German-Israeli Foundation for Research and Development and by the Hermann-Minkowski Minerva Center for Geometry at Tel Aviv University. This work was completed during the author’s RiP stay at the Mathematisches Forschunsinstitut Oberwolfach.
Article copyright: © Copyright 2003 American Mathematical Society