Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem

Christophe Eyral; Elizabeth Gasparim

doi:10.1016/j.crma.2007.04.005

Algebraic Geometry

Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem

Presented by: Bernard Malgrange

Christophe Eyral ¹; Elizabeth Gasparim ²

¹ Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
² The University of Edinburgh, School of Mathematics, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 631-634.

Abstract
Résumé

Let $f, g : (C^{n}, 0) \to (C, 0)$ be reduced germs of holomorphic functions. We show that f and g have the same multiplicity at 0, if and only if, there exist reduced germs $f^{'}$ and $g^{'}$ analytically equivalent to f and g, respectively, such that $f^{'}$ and $g^{'}$ satisfy a Rouché type inequality with respect to a generic ‘small’ circle around 0. As an application, we give a reformulation of Zariski's multiplicity question and a partial positive answer to it.

Soient $f, g : (C^{n}, 0) \to (C, 0)$ des germes de fonctions holomorphes réduits. Nous montrons que f et g ont la même multiplicité en 0 si et seulement s'il existe des germes réduits $f^{'}$ et $g^{'}$ analytiquement équivalents à f et g, respectivement, tels que $f^{'}$ et $g^{'}$ satisfassent une inégalité du type de Rouché par rapport à un ‘petit’ cercle générique autour de 0. Comme application, nous donnons une reformulation de la question de Zariski sur la multiplicité et une réponse partielle positive à celle-ci.

Received: 2006-12-04
Accepted: 2007-04-03
Published online: 2007-05-15

DOI: 10.1016/j.crma.2007.04.005

Author's affiliations:

Christophe Eyral ¹; Elizabeth Gasparim ²

¹ Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
² The University of Edinburgh, School of Mathematics, James Clerk Maxwell Building, The King's Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom

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Christophe Eyral; Elizabeth Gasparim. Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem. Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 631-634. doi : 10.1016/j.crma.2007.04.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.005/

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Cited by Sources:

^⁎ This research was supported by the Max-Planck Institut für Mathematik in Bonn.

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