Abstract

A collection of n+1 real functions on an interval $I$ is called a Chebyshev system if no nontrivial linear combination of them has more than n isolated zeros on I counted with multiplicities. In many situations it is necessary to show that a collection of Abelian integrals is a Chebyshev system. Abelian integrals appear naturally when studying bifurcations of limit cycles of planar polynomial vector fields. In particular, zeros of Abelian integrals are related to limit cycles appearing in polynomial perturbations of polynomial Hamiltonian vector fields.
In general it is very difficult to show that a given collection of Abelian integrals is a Chebyshev system. The traditional approach is to use properties of the system of linear ordinary differential equations satisfied by the Abelian integrals, the so-called Picard-Fuchs system. However, sometimes, even the effective computation of this system turns out to be a difficult problem. In this talk I will explain a very simple condition that guarantees that a collection of Abelian integrals is Chebyshev.