Abstract

Let f, g : (C-n, 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 Rouche 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. To cite this article: C. Eyral, E. Gasparim, C. R. Acad. Sci. Paris, Ser. 1 344 (2007). (c) 2007 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.