On the number of zeros of Abelian integrals for discontinuous quadratic differential systems

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all discontinuous polynomials with degree $n$. Furthermore, by using the {\it Chebyshev criterion}, we obtain the sharp upper bounds on each period annulus for $n=2$.