Maximal smoothings of real plane curve singular points

To V.A.Rokhlin who guided certain of us in the marvelous world behind reals. Abstract 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 real singular points for which this bound is not sharp. We show that Harnack inequality is sharp in any complex topologically equisingular class: every real 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 reened Harnack inequality and its sharpness are discussed as well. This work was started during the stay of the third author at Ecole Normale Sup erieure and Univerist e de Strasbourg, continued during the stay of the rst and third author at Universitt at Kaiserslautern and nished during the stay of the third author at the University of Toronto. We thank all these institutions for hospitality and nancial support.