ISRAEL JOURNAL OF MATHEMATICS 125 (2001), 293-315 EQUIMULTIPLE DEFORMATIONS OF ISOLATED SINGULARITIES BY I. SCHERBACK AND E. SHUSTIN* School of Mathematical Sciences, Tel Aviv University Ramat Aviv, Tel Aviv 69978, Israel e-mail: scherbak@post.tau.ac.il, shustin@post.tau.ac.il ABSTRACT We study equimultiple deformations of isolated hypersurface singularities, introduce a blow-up equivalence of singular points, which is intermedi- ate between topological and analytic ones, and give numerical sufficient conditions for the blow-up versality of the equimultiple deformation of a singularity or multisingularity induced by the space of algebraic hyper- surfaces of a given degree. For singular points, which become Newton nondegenerate after one blowing up, we prove that the space of algebraic hypersurfaces of a given degree induces all the equimultiple deformations (up to the blow-up equivalence) which are stable with respect to removing monomials lying above the Newton diagrams. This is a generalization of a theorem by B. Chevallier. Introduction This paper is devoted to deformations of a special kind for isolated hypersurface singularities over the real or complex field. A classical problem is to describe what happens with singularities of an alge- braic hypersurface when varying in the space of hypersurfaces of a given degree (or in another interesting class). The versality of a deformation guarantees that it contains all possible deformations in the considered class. The case of algebraic * This work was partially supported by Grant No.6836-1-9 of the Israeli Ministry of Sciences. The second author thanks the Max-Planck Institut (Bonn) for hos- pitality and financial support. Received June 21, 2000 293