Multi-Harnack smoothings of real plane branches
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 1, pp. 143-184.

Let Δ𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion of Harnack curves, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, via Viro's patchworking method, and that the topological type of their real parts is unique (and determined by Δ). 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 of multi-Harnack smoothings of (C,0) by passing through a resolution of singularities of (C,0) consisting of g monomial maps (where g is the number of characteristic pairs of the branch). A multi-Harnack smoothing is a g-parametrical deformation which arises as the result of a sequence, beginning at the last step of the resolution, consisting of a suitable Harnack smoothing (in terms of Mikhalkin's definition) followed by the corresponding monomial blow down. We prove then the unicity of the topological type of a multi-Harnack smoothing. In addition, the multi-Harnack smoothings can be seen as multi-semi-quasi-homogeneous in terms of the parameters. Using this property we analyze the asymptotic multi-scales of the ovals of a multi-Harnack smoothing. We prove that these scales characterize and are characterized by the equisingularity class of the branch.

Soit Δ𝐑 2 un polygone convexe à sommets entiers ; G. Mikhalkin a défini les « courbes de Harnack » (définies par un polynôme de support contenu dans Δ et plongées dans la surface torique correspondante) et montré leur existence (via la « méthode du patchwork de Viro ») ainsi que l’unicité de leur type topologique plongé (qui est determiné par Δ). Le but de cet article est de montrer un résultat analogue pour la lissification (smoothing) d’un germe de branche réelle plane (C,O) analytique réelle. On définit pour cela une classe de smoothings dite « Multi-Harnack » à l’aide de la résolution des singularités constituée d’une suite de g éclatements toriques, si g est le nombre de paires de Puiseux de la branche (C,O). Un smoothing multi-Harnack est réalisé de la manière suivante : à chaque étape de la résolution (en commençant par la dernière) et de manière successive, un smoothing « De Harnack » (au sens de Mikhalkin) intermédiaire est obtenu par la méthode de Viro. On montre alors l'unicité du type topologique de tels smoothings. De plus, on peut supposer ces smoothings « multi-semi-quasi homogènes »  ; on montre alors que des propriétés métriques (« multi-taille » des ovales) de tels smoothings sont caractérisées en fonction de la classe d’équisingularité de (C,O) et que réciproquement ces tailles caractérisent la classe d’équisingularité de la branche.

DOI: 10.24033/asens.2118
Classification: 14P25,  14H20,  14M25
Keywords: smoothings of singularities, real algebraic curves, harnack curves
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     title = {Multi-Harnack smoothings of real plane branches},
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González Pérez, Pedro Daniel; Risler, Jean-Jacques. Multi-Harnack smoothings of real plane branches. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 1, pp. 143-184. doi : 10.24033/asens.2118. http://www.numdam.org/articles/10.24033/asens.2118/

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