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RESEARCH PRODUCT

Pseudo-abelian integrals: Unfolding generic exponential case

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The search for bounds on the number of zeroes of Abelian integrals is motivated, for instance, by a weak version of Hilbert's 16th problem (second part). In that case one considers planar polynomial Hamiltonian perturbations of a suitable polynomial Hamiltonian system, having a closed separatrix bounding an area filled by closed orbits and an equilibrium. Abelian integrals arise as the first derivative of the displacement function with respect to the energy level. The existence of a bound on the number of zeroes of these integrals has been obtained by A. N. Varchenko [Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 14–25 ; and A. G. Khovanskii [Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 40–50. The present article is devoted to the search for the extension of the result of Varchenko and Khovanskii in the case of a (generalized) Darboux integrable planar system. Hence the unperturbed system is the real closed $1$-form $$ \theta_0 = d \log H_0, $$ where $H_0 = P_1^{; ; a_1}; ; \cdots P_k^{; ; a_k}; ; e^{; ; RQ}; ; $, and the unfolding $\theta_{; ; \varepsilon , \alpha}; ; $ of the unperturbed system is (generalized) Darboux integrable and has a maximal nest of cycles each contained in the $h$-level of the integral, $h \in (0, b(\varepsilon , \alpha))$. Then pseudo-Abelian integrals $I_{; ; \varepsilon , \alpha}; ; $ appear as the linear term with respect to $\delta$ of the displacement function of a polynomial deformation $$ M \theta_{; ; \varepsilon , \alpha}; ; + \delta \eta, $$ where $\eta$ is a polynomial $1$-form of degree $n$ and $M= {; ; Q(Q+\varepsilon R)P_1 \cdots P_k}; ; $. The main result in this article is the existence of a bound for the number of isolated zeroes of the pseudo-Abelian integrals$I_{; ; \varepsilon , \alpha}; ; $ under the genericity hypothesis that the curves $P_J^{; ; -1}; ; (0)$ and $Q^{; ; -1}; ; (0)$ are smooth and intersect transversally.