Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x^′=−yG(x,y),y^′=xG(x,y)$x^{\prime }=-yG(x,y),y^{\prime }=xG(x,y)$, where the set of critical points {G(x,y)=0}$\{G(x,y)=0\}$ is formed by K$K$ 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 n$n$ and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K$K$ and n$n$. 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≤4$K\le 4$ we recover or improve some results obtained in several previous works.