Bifurcation of limit cycles from a polynomial degenerate center

Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center x = -y((x2 + y2)/2)m and y = x((x2 + y2)/2)m with m ≥ 1, when we perturb it inside the whole class of polynomial vector fields of degree n.The positive integers m and n are arbitrary. As far as we know there is only one paper that provide a similar result working with Melnikov functions at any order and perturbing the linear center x = - y, y = x.