Estimate of the number of zeros of Abelian integrals for an elliptic Hamiltonian with figure-of-eight loop

Abstract

We study upper bounds of the number of zeros of Abelian integrals of polynomial 1-forms of degree n over the compact level curves Γh of the elliptic Hamiltonian of degree four, having a `figure-of-eight loop'. Upper bounds are given for Γh lying inside the figure-of-eight loop and for Γh lying outside the figure-of-eight loop. The bounds are linearly dependent on n.