On the Chebyshev Property of Certain Abelian Integrals Near a Polycycle

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Dumortier and Roussarie formulated in (Discrete Contin Dyn Syst 2:723–781, 2009) a conjecture concerning the Chebyshev property of a collection \(I_0,I_1,\ldots ,I_n\) of Abelian integrals arising from singular perturbation problems occurring in planar slow-fast systems. The aim of this note is to show the validity of this conjecture near the polycycle at the boundary of the family of ovals defining the Abelian integrals. As a corollary of this local result we get that the linear span \(\langle I_0,I_1,\ldots ,I_n \rangle \) is Chebyshev with accuracy \(k=k(n)\).