Upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one

In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we study the upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one under any polynomial perturbations of degree $n$, and obtain that their upper bounds are $3n-3$ ($ngeq 2$) and $18left[frac{n}{2}right]+3left[frac{n-1}{2}right]$ ($ngeq 4$) respectively, both of the two upper bounds linearly depend on $n$.