Algebraic Geometry
Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem
Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 631-634.

Let $f,g:(Cn,0)→(C,0)$ 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 $f,g:(Cn,0)→(C,0)$ 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.04.005

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.