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

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 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. To cite this article: C. Eyral, E. Gasparim, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Enumeration of uni-singular algebraic hypersurfaces

“…To achieve the Newton diagram of A 4 we must do the non-linear shift: z i → z i + δ i z 2 1 (to kill the monomials z 4 2 , z 2 1 z i ). Elimination of the parameters of the transformation gives the non-linear equation γ = β 2 i /(4α i ).…”

“…The use of adjacency: tacnodal hypersurfaces. The tacnodal singularity (A 3 ) has the normal form f = z 4 1 + n i=2 x 2 i . The corresponding lifted variety was defined in Example 3.4.…”

“…The associated Newton diagram can be non-constant along the equisingular stratum (in the sense that the transformation needed to achieve it, is not locally analytic but a homeomorphism) [Dimca, chapter 1, example 2.14]. The constancy of multiplicity along the equisingular stratum has up to now been proved for semi-quasi-homogeneous singularities only [Greue86,GrePfi96] (see also [EyrGas05] for recent results).…”

On Zariski's multiplicity conjecture

“…Here we discuss some features of the problem, especially the relations of the work of A'Campo on the zeta function of a monodromy and the Zariski's multiplicity conjecture. Also some previous results are sharpened; the results of [6] and [18] in theorem (3.2) and the one in [7] in theorem (5.2). In an analogy with hypersurfaces, J.F.…”