# Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez; Jean-Jacques Risler

Annales scientifiques de l'École Normale Supérieure (2010)

- Volume: 43, Issue: 1, page 143-184
- ISSN: 0012-9593

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topGonzález Pérez, Pedro Daniel, and Risler, Jean-Jacques. "Multi-Harnack smoothings of real plane branches." Annales scientifiques de l'École Normale Supérieure 43.1 (2010): 143-184. <http://eudml.org/doc/272195>.

@article{GonzálezPérez2010,

abstract = {Let $\Delta \subset \mathbf \{R\}^2$ be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on $\Delta $ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, 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 ofmulti-Harnacksmoothings 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 suitableHarnack 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.},

author = {González Pérez, Pedro Daniel, Risler, Jean-Jacques},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {smoothings of singularities; real algebraic curves; harnack curves},

language = {eng},

number = {1},

pages = {143-184},

publisher = {Société mathématique de France},

title = {Multi-Harnack smoothings of real plane branches},

url = {http://eudml.org/doc/272195},

volume = {43},

year = {2010},

}

TY - JOUR

AU - González Pérez, Pedro Daniel

AU - Risler, Jean-Jacques

TI - Multi-Harnack smoothings of real plane branches

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2010

PB - Société mathématique de France

VL - 43

IS - 1

SP - 143

EP - 184

AB - Let $\Delta \subset \mathbf {R}^2$ be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on $\Delta $ and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, 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 ofmulti-Harnacksmoothings 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 suitableHarnack 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.

LA - eng

KW - smoothings of singularities; real algebraic curves; harnack curves

UR - http://eudml.org/doc/272195

ER -

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