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Aymen Braghtha

Publications and source records attributed to Aymen Braghtha.

4 recordsLinked to original sources
arXiv2016

Darboux integrable system with a triple point and pseudo-abelian integrals

In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves $\{P_i = 0\}$ meet at one point, are considered. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for close Darboux integrable foliations. Let $\mathcal{F}$ denote the foliation with triple point (assume it to be at the origin), and let $\mathcal{F}_λ= \{M_λ{dH_λ\over H_λ} = 0\}$, $M_λ$ is a integrating factor, be the close foliation. The main problem is that $\mathcal{F}_λ$ can have a small nest of cycles which shrinks to the origin as $λ\to 0$. A particular case of this situation, namely $H_λ= (x -λ)^ε(y - x)^{ε_+} (y + x)^{ε_-}Δ$ with $Δ$ non-vanishing at the origin (and generic in appropriate sense).

math.DS↗
arXiv2015

On the Number of Isolated Zeros of Pseudo-Abelian Integrals: Degeneracies of the Cuspidal Type

We consider a multivalued function of the form $H\_{\varepsilon}=P\_{\varepsilon}^{α\_0}\prod^{k}\_{i=1}P\_i^{α\_i}, P\_i\in\mathbb{R}[x,y], α\_i\in\mathbb{R}^{\ast}\_+$, which is a Darboux first integral of polynomial one-form $ω=M\_{\varepsilon}\frac{dH\_{\varepsilon}}{H\_{\varepsilon}}=0, M\_{\varepsilon}=P\_{\varepsilon}\prod^{k}\_{i=1}P\_i$. 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.

math.DS↗
arXiv2015

Local monomialization conjecture of a singular foliation of Darboux type

After the nice result introduced by Belotto in [1] concerning the local monomialization of a singular foliation given by n first integrals, this work is a continuation in the same spirit. In this paper, we introduce a important conjecture about local monomialization of a singular foliation of Darboux type (see section 1). This conjecture can be used to study pseudo-abelian integrals [2,4].

math.DS↗