Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field $x^{\mathrm{\prime }}=-yG(x,y),y^{\mathrm{\prime }}=xG(x,y),$ where the set of critical points $\{G(x,y)=0\}$ is formed by $K$ 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 $n$ and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of $K$ and $n\mathrm{.}$ Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for $K\le 4$ we recover or improve some results obtained in several previous works.