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González Pérez, Pedro Daniel and Risler, JeanJacques (2010) MultiHarnack smoothings of real plane branches. Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 43 (1). pp. 143183. ISSN 00129593

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Official URL: http://www.numdam.org/item/ASENS_2010_4_43_1_143_0
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 multiHarnack smoothings of (C, 0) and prove its existence along with its topological uniqueness.
Theorem 9.3. Any real plane branch (C, 0) has a multiHarnack smoothing.
Theorem 9.4. Let (C, 0) be a real branch. The topological type of multiHarnack smoothings of (C, 0) is unique. There are at most two signed topological types of multiHarnack 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, multiHarnack smoothings are multisemiquasihomogeneous, which lets the authors analyze also the asymptotic multiscales 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:  14 May 2018 11:18 
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