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

Consider planar ordinary differential equations of the form x=-yC(x,y),y=xC(x,y), where C(x,y) is an algebraic curve. We are interested in knowing whether the existence of multiple factors for C is important or not when we study the maximum number of zeros of the Abelian integral M 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)m, where m is a positive integer number and prove that m has essentially no impact on the number of zeros of M. This result improves the known studies on M. One of the key points of our approach is that we obtain a simple expression of M based on some successive reductions of the integrals appearing during the procedure. © 2011 Elsevier Ltd. All rights reserved.