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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.04.005
Eyral, Christophe 1; Gasparim, Elizabeth 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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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. http://www.numdam.org/articles/10.1016/j.crma.2007.04.005/

[1] O.M. Abderrahmane, On the deformation with constant Milnor number and Newton polyhedron, Preprint, Saitama University, 2004

[2] Ephraim, R. C1 preservation of the multiplicity, Duke Math. J., Volume 43 (1976), pp. 797-803

[3] C. Eyral, Zariski's multiplicity question – a survey, Preprint, Max-Planck Institut für Mathematik, 2005

[4] King, H. Topological type of isolated critical points, Ann. of Math., Volume 107 (1978), pp. 385-397

[5] King, H. Topological type in families of germs, Invent. Math., Volume 62 (1980), pp. 1-13

[6] Kouchnirenko, A.G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976), pp. 1-32

[7] Lang, S. Complex Analysis, Graduate Texts in Mathematics, vol. 103, Springer–Verlag, 1995

[8] T. Nishimura, A remark on topological types of complex isolated singularities of hypersurfaces, private communication between O. Saeki and T. Nishimura as cited in [11]

[9] Oka, M. Non-Degenerate Complete Intersection Singularity, Actualités Mathématiques, Hermann, Paris, 1997

[10] Perron, B. Conjugaison topologique des germes de fonctions holomorphes à singularité isolée en dimension trois, Invent. Math., Volume 82 (1985), pp. 27-35

[11] Saeki, O. Topological types of complex isolated hypersurface singularities, Kodai Math. J., Volume 12 (1989), pp. 23-29

[12] Saia, M.J.; Tomazella, J.N. Deformations with constant Milnor number and multiplicity of complex hypersurfaces, Glasg. Math. J., Volume 46 (2004), pp. 121-130

[13] Whitney, H. Complex Analytic Varieties, Addison-Wesley Publishing Company, Reading, MA–Menlo Park, CA, 1972

[14] Zariski, O. Open questions in the theory of singularities, Bull. Amer. Math. Soc., Volume 77 (1971), pp. 481-491

Cited by Sources:

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