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Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization

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  • We discuss polynomial 1-form small perturbation of an isochronous polynomial 1-form in the Pfaffian form $\omega_{\epsilon}=\omega_0+\epsilon \omega$ where $\omega$ is a n-degree polynomial 1-form, $\epsilon$ a small real parameter, and $\omega_0$ an isochronous 1-form with a known birational linearization $T,$ setting $\omega_0$ as the pullback 1-form $T^*\Cal I_0$ of the exact linear isochrone 1-form $\Cal I_0=dH.$ Using recursively the cohomology decompositions of $\omega$ in the related Petrov module, we construct the Bautin-like ideal of the Poincar\'e-Melnikov functions, and study the zeros of Abelian integrals over the ovals $\tilde H=T^*H=r.$ We then stabilize the sequence of the successive Melnikov functions through a multistep reduction of the system coefficients, and determine in terms of the degrees of $\tilde H$ and $\omega$ the overall upper bounds for limit cycles emerging from the polynomial deformation.
    Mathematics Subject Classification: 58F14, 58F21, 34C15, 34C25.

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