### Multi-Harnack smoothings of real plane branches

Let $\Delta \subset {\mathbf{R}}^{2}$ be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on $\Delta $ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by $\Delta $). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch $(C,0)$. We introduce the class ofsmoothings of $(C,0)$ by passing through...