Abstract

Given a two-dimensional Hamiltonian system, we can calculate the number of limit cycles of its first-order perturbations via a family of Abelian integrals. When these vanish identically, then it is possible to consider second order terms, but are there also geometric consequences to the vanishing of these integrals? In this talk we consider the case of a hyperelliptic Hamiltonian, and show that those Abelian integrals which are not relatively exact have Hamiltonians which are decomposible or are reducible to a small class of Hamiltonians based on Chebyshev polynomial.