Abstract

F. Dumortier and R. Roussarie formulated in [Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009), 723-781] 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 talk 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 $langle I_0,I_1,ldots,I_n angle$ is Chebyshev with accuracy $k=k(n).$

This result is a joint work with David Marín.