The Monodromy Problem and the Tangential Center Problem

We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center–focus problem asks for conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both of these questions for the case in which the Hamiltonian is hyperelliptic. As a by-product, we solve the corresponding problems for the $0$-dimensional Abelian integrals defined by Gavrilov and Movasati.