# 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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https://hal.archives-ouvertes.fr/hal-01145681
Contributor : Aymen Braghtha <>
Submitted on : Saturday, April 25, 2015 - 1:34:19 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
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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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