Melnikov Functions Appearing in Polynomial Hamiltonian Perturbations

Joint project with Pavao Mardesic, Laura Ortiz-Bobadilla, and Jessie Pontigo-Herrera.

Hibert's 16th problem asks for an upper bound on the number of limit cycles of planar polynomial vector fields. For polynomial perturbations \dH+ϵω

of planar polynomial foliations, this is closely related to isolated zeros of the Abelian integrals ∫δω.

However, in degenerate cases, the first-order approximation given by Abelian integrals vanishes, and one should consider higher-order approximations given by Chen's iterated integrals like ∫δωω′. We are trying to understand their finiteness properties, which are closely related to the monodromy orbit of δ in π1({H=t}).