González Pérez, Pedro DanielRisler, Jean-Jacques2023-06-202023-06-2020100012-959310.24033/asens.2118https://hdl.handle.net/20.500.14352/41939This paper is motivated by the results of G. Mikhalkin about a certain class of real algebraic curves, called Harnack curves, in toric surfaces. Mikhalkin has proved the existence of such curves as well as topological uniqueness of their real locus. The authors are concerned about an analogous statement in the case of a smoothing of a real plane branch (C, 0) _ (C2, 0) (an analytically irreducible germ of a real curve). They introduce the class of multi-Harnack smoothings of (C, 0) and prove its existence along with its topological uniqueness. Theorem 9.3. Any real plane branch (C, 0) has a multi-Harnack smoothing. Theorem 9.4. Let (C, 0) be a real branch. The topological type of multi-Harnack smoothings of (C, 0) is unique. There are at most two signed topological types of multi-Harnack smoothings of (C, 0). These types depend only on the sequence {(nj ,mj)}, which determines and is determined by the embedded topological type of (C, 0) _ (C2, 0). In terms of the parameters, multi-Harnack smoothings are multi-semi-quasi-homogeneous, which lets the authors analyze also the asymptotic multi-scales of the ovals.engjournal articlehttp://www.numdam.org/item/ASENS_2010_4_43_1_143_0open access512.7Smoothings of singularitiesReal algebraic curvesHarnack curvesGeometria algebraica1201.01 Geometría Algebraica