An improved estimate for the number of zeros of Abelian integrals for cubic Hamiltonians

Suppose that the real generic cubic Hamiltonian H(x, y), (x,y) element of R2, possesses three saddle points and one centre. Let Σ subset of R be the set of values h of H(x, y), for which there exists a closed component δ(h) of the level curve {H(x, y) = h}, free of critical points. In this paper, we obtain a better upper bound than previously known for the number of zeros of the Abelian integrals I(h) = ∫δ(h)[g(x, y) dx − f(x, y) dy] for h in Σ in terms of the maximum of the degrees of the polynomials f(x, y) and g(x, y)