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Multi-Harnack smoothings of real plane branches

González Pérez, Pedro Daniel and Risler, Jean-Jacques (2010) Multi-Harnack smoothings of real plane branches. Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 43 (1). pp. 143-183. ISSN 0012-9593

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Abstract

This 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.


Item Type:Article
Uncontrolled Keywords:Smoothings of singularities, Real algebraic curves, Harnack curves
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:12719
Deposited On:20 May 2011 11:36
Last Modified:06 Feb 2014 09:31

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