Artal Bartolo, E. et al. (2010) “Milnor Number of Weighted-Lˆ e–Yomdin Singularities,” International Mathematics Research Notices, Vol. 2010, No. 22, pp. 4301–4318 Advance Access publication March 14, 2010 doi:10.1093/imrn/rnq041 Milnor Number of Weighted-L ˆ e–Yomdin Singularities E. Artal Bartolo 1 , J. Fern ´ andez de Bobadilla 2,* , I. Luengo 3 , and A. Melle-Hern ´ andez 3 1 Departamento de Matem ´ aticas, Facultad de Ciencias, IUMA, Universidad de Zaragoza, c/Pedro Cerbuna 12, E-50009 Zaragoza, Spain, 2 ICMAT, CSIC-Complutense-Aut´ onoma-Carlos III, Madrid, Spain, and 3 Departamento de ´ Algebra, Universidad Complutense, Plaza de las Ciencias s/n, Ciudad Universitaria, 28040 Madrid, Spain Correspondence to be sent to: javier@mat.csic.es At the beginning of the seventies, O. Zariski proposed several problems related with the (embedded) topology of a germ of a n -dimensional hypersurface singularity de- fined by the zero locus of a germ of a complex analytic function. The second one was roughly stated as “if two analytic hypersurface germs are topologically equivalent then their tangent cones must be homeomorphic and the homeomorphism must respect the topological equisingularity type at any point.” In this paper, we give counter- examples for n = 3 and 4 (even in a family). Our proof is mainly based on the study of the topology of weighted-L ˆ e–Yomdin surface singularities which are a generalization of the well-known Lˆ e–Yomdin singularities. We obtain a formula for the Milnor number of a weighted-L ˆ e–Yomdin surface singularity and derive an equisingularity criterion for them. In [19], Zariski proposed to study a series of problems (from A to H) related with the (embedded) topology of a germ of a hypersurface singularity (V, 0) ⊂ (C n , 0) defined by the zero locus of a germ of a complex analytic function f : (C n , 0) → (C, 0). He defined Received October 19, 2009; Revised January 28, 2010; Accepted February 15, 2010 Communicated by Prof. Thomas Bloom c  The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.