Let A c R2 be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on A and containedin the corresponding toric surface. He proved their existence, via Viro's patchworking method, and thatthe topological type of their real parts is unique (and determined by A). This paper is concerned withthe 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 sin-gularities of (C, 0) consisting of g monomial maps (where g is the number of characteristic pairs ofthe branch). A multi-Harnack smoothing is a g-parametrical deformation which arises as the resultof 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 provethen the unicity of the topological type of a multi-Harnack smoothing. In addition, the multi-Harnacksmoothings can be seen as multi-semi-quasi-homogeneous in terms of the parameters. Using this prop-erty we analyze the asymptotic multi-scales of the ovals of a multi-Harnack smoothing. We prove thatthese scales characterize and are characterized by the equisingularity class of the branch.
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机译：设A c R2为整数凸多边形。 G. Mikhalkin引入了Harnack曲线的概念，Harnack曲线是一类实数代数曲线，由A上支持的多项式定义并包含在对应的复曲面中。他通过Viro的拼凑方法证明了它们的存在，并且它们的实际部分的拓扑类型是唯一的（由A确定）。本文关注对实平面分支（C，0）进行平滑处理时的类似陈述的描述。我们通过对sin-的解析来介绍（C，0）的多Harnack平滑类别。 （C，0）的凸度由g个单项式映射组成（其中g是分支的特征对的数量）。多重Harnack平滑是一个g参数变形，它是从分辨率的最后一步开始的一系列结果产生的，该序列包括一个合适的Harnack平滑（根据Mikhalkin的定义），然后是对应的单项式分解。我们证明了多重哈纳克平滑的拓扑类型的唯一性。另外，就参数而言，多Harnacks平滑可以看作是多半准同质的。使用此属性，我们分析了多哈纳克平滑的椭圆的渐近多尺度。我们证明了这些量表的特征在于分支的等价性类。
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