Let 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 and analytically equivalent to f and g, respectively, such that and 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 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 et analytiquement équivalents à f et g, respectivement, tels que et 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:
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@article{CRMATH_2007__344_10_631_0,
author = {Eyral, Christophe and Gasparim, Elizabeth},
title = {Multiplicity of complex hypersurface singularities, {Rouch\'e} satellites and {Zariski's} problem},
journal = {Comptes Rendus. Math\'ematique},
pages = {631--634},
publisher = {Elsevier},
volume = {344},
number = {10},
year = {2007},
doi = {10.1016/j.crma.2007.04.005},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.crma.2007.04.005/}
}
TY - JOUR AU - Eyral, Christophe AU - Gasparim, Elizabeth TI - Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem JO - Comptes Rendus. Mathématique PY - 2007 SP - 631 EP - 634 VL - 344 IS - 10 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.crma.2007.04.005/ DO - 10.1016/j.crma.2007.04.005 LA - en ID - CRMATH_2007__344_10_631_0 ER -
%0 Journal Article %A Eyral, Christophe %A Gasparim, Elizabeth %T Multiplicity of complex hypersurface singularities, Rouché satellites and Zariski's problem %J Comptes Rendus. Mathématique %D 2007 %P 631-634 %V 344 %N 10 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.crma.2007.04.005/ %R 10.1016/j.crma.2007.04.005 %G en %F CRMATH_2007__344_10_631_0
Eyral, Christophe; Gasparim, Elizabeth. 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://www.numdam.org/articles/10.1016/j.crma.2007.04.005/
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⁎ This research was supported by the Max-Planck Institut für Mathematik in Bonn.
