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 (1 + x) dH = 0, where H is the quasi-homogeneous Hamiltonian H(x, y) = x(2k)/(2k) + y(2)/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. (c) 2006 Elsevier Inc. All rights reserved.