Singularity Which Has No M-Smoothing

. The Harnack bound on the number of real components of a plane real algebraic curve has a natural local version which states that the number of closed real components obtained by a perturbation of a real isolated plane curve singularity having at least one real branch is bounded by the genus of the singularity (perturbations attending this extremal value are called M-smoothings). We show that the latter bound is not sharp for some, explicitly given, singularities. Introduction Topologically extremal real algebraic varieties reveal spectacular topological properties. Such a phenomenon for plane projective curves was discovered by D. Hilbert, and is stated in the first part of his 16th problem in the form of conjecture for plane curves of degree 6 with maximal number of real components. D.A. Gudkov [6] corrected the conjecture, generalized it to curves of arbitrary degree and proved it for degree 6 by the meticulous study of degenerations of curves of degree 6 (improving the so-called ..