Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x′=−yG(x,y),y′=xG(x,y)x′=−yG(x,y),y′=xG(x,y), where the set of critical points {G(x,y)=0}{G(x,y)=0} is formed by KK straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree nn and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of KK and nn. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4K≤4 we recover or improve some results obtained in several previous works.