Maximal smoothings of real plane curve singular points.

The local Harnack inequality bounds from above the number of ovals which can appear in a small perturbation of a singular point. As is known, there are singular points for which this bound is not sharp. We show that Harnack inequality is sharp in any complex topologically equisingular class: every real plane curve singular point is complex deformation equivalent to a real singularity for which Harnack inequality is sharp. For semi-quasi-homogeneous and some other singularities we exhibit a real deformation with the same property. A refined Harnack inequality and its sharpness are discussed also