Limit cycles bifurcating from the period annulus of quasi-homogeneous centers (Q1016059)

The paper deals with the weakened 16th Hilbert's problem. It contains upper bounds for the maximum number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous center of the perturbed Hamiltonian system \[ \dot{x}=-\frac{\partial H}{\partial y} + \varepsilon P, \;\dot{y}=\frac{\partial H}{\partial x} + \varepsilon Q, \] where \(H\), \(P\) and \(Q\) are polynomials in \((x,y)\) and \(\varepsilon \geq 0\) is a small real parameter. The authors show that these bounds are the best possible using the Abelian integral method of first order and provide the biggest known number of limit cycles surrounding a unique equilibria in terms of the degree \(n\) of the system.