Limit cycles appearing from the perturbation of a system with a multiple line of critical points

Consider planar ordinary differential equations of the form ẋ=−yC(x,y),ẏ=xC(x,y), where C(x,y)C(x,y) is an algebraic curve. We are interested in knowing whether the existence of multiple factors for CC is important or not when we study the maximum number of zeros of the Abelian integral MM that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field. With this aim, we study in detail the case C(x,y)=(1−y)mC(x,y)=(1−y)m, where mm is a positive integer number and prove that mm has essentially no impact on the number of zeros of MM. This result improves the known studies on MM. One of the key points of our approach is that we obtain a simple expression of MM based on some successive reductions of the integrals appearing during the procedure.