ON THE NUMBER OF ISOLATED ZEROS OF PSEUDO-ABELIAN INTEGRALS: DEGENERACIES OF THE CUSPIDAL TYPE

Abstract : We consider a multivalued function of the form $H_{\varepsilon}=P_{\varepsilon}^{\alpha_0}\prod^{k}_{i=1}P_i^{\alpha_i}, P_i\in\mathbb{R}[x,y], \alpha_i\in\mathbb{R}^{\ast}_+$, which is a Darboux first integral of polynomial one-form $\omega=M_{\varepsilon}\frac{dH_{\varepsilon}}{H_{\varepsilon}}=0, M_{\varepsilon}=P_{\varepsilon}\prod^{k}_{i=1}P_i$. We assume, for $\varepsilon=0$, that the polycyle $\{H_0=H=0\}$ has only cuspidal singularity which we assume at the origin and other singularities are saddles. We consider families of Darboux first integrals unfolding $H_{\varepsilon}$ (and its cuspidal point) and pseudo-Abelian integrals associated to these unfolding. Under some conditions we show the existence of uniform local bound for the number of zeros of these pseudo-Abelian integrals.
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Type de document :
Pré-publication, Document de travail
2015

https://hal.archives-ouvertes.fr/hal-01145681
Contributeur : Aymen Braghtha <>
Soumis le : samedi 25 avril 2015 - 13:34:19
Dernière modification le : lundi 21 mars 2016 - 11:32:32
Document(s) archivé(s) le : lundi 14 septembre 2015 - 13:26:58

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• HAL Id : hal-01145681, version 1
• ARXIV : 1506.05964

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Aymen Braghtha. ON THE NUMBER OF ISOLATED ZEROS OF PSEUDO-ABELIAN INTEGRALS: DEGENERACIES OF THE CUSPIDAL TYPE. 2015. <hal-01145681>

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