Jordi Villadelprat, (UB): A Chebyshev criterion for Abelian integrals

Let E be an n-dimensional linear space of analytic functions on a
real interval I. E is said to be an extended Chebyshev space if any
nonzero element of E has at most n-1 zeros on I counted with
multiplicities. Abelian integrals appear naturally when studying
planar differential systems. For instance, the first approximation
of the displacement function of the Poincaré map associated to a
small deformation of a Hamiltonian system is an Abelian integral.
Thus, under generic conditions, its zeros will determine the number
and location of limit cycles born in the perturbation.

Bounding the number of zeros of an Abelian integral is usually a
very long and highly non-trivial problem. In some papers the authors
study the geometrical properties of the so-called centroid curve
using the fact that it verifies a Riccati equation (which is itself
deduced from a Picard-Fuchs system). In other papers the authors use
complex analysis and algebraic topology (analytic continuation,
argument principle, monodromy, Picard-Lefschetz formula, etc.).

In this talk I will explain a very simple condition that guarantees
that a collection of Abelian integrals is Chebyshev. This condition
involves the functions in the integrand of the Abelian integrals and
can be checked, in many cases, in a purely algebraic way. By using
this criterion, several known results are obtained in a shorter way
and some new results, which could not be tackled by the known
standard methods, can also be deduced.

This is a joint work with Maite Grau(UdL) and Francesc Mañosas(UAB)