Monodromy and Tangential Center Problems

TL;DR

This paper investigates conditions for vanishing Abelian integrals in perturbed Hamiltonian systems, linking monodromy and tangential center problems, and provides solutions specifically for hyperelliptic Hamiltonians and 0-dimensional integrals.

Contribution

It offers new solutions to the monodromy and tangential center problems for hyperelliptic Hamiltonians and extends results to 0-dimensional Abelian integrals.

Findings

Solved tangential center conditions for hyperelliptic Hamiltonians

Established monodromy generation criteria for these systems

Extended results to 0-dimensional Abelian integrals

Abstract

We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center focus problem asks for the 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 these questions for the case when the Hamiltonian is hyperelliptic. As a side-product, we solve the corresponding problems for the "0-dimensional Abelian integrals" defined by Gavrilov and Movasati.