On the number of limit cycles bifurcating from a non-global degenerated center

We give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree n, from the periodic orbits of the integrable system , where H is the quasi-homogeneous Hamiltonian H(x,y)=x2k/(2k)+y2/2. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis.