Abstract

For any given set of real continuous functions F={g_0(x), g_1(x), . . . , g_n (x)}, the study of how many zeroes have any linear combination of functions of F is, in general, very difficult. The Chebyshev properties of such families solve the problem, we will show how to do that for a concrete family. The number of zeroes of some abelian integrals can be computed using this theory. We will apply the results to some particular families of perturbations of polynomial vector fields in the plane.

This is a joint work with A. Gasull and J.T. Lázaro.