On the Number of Zeros of Abelian Integrals for a Class of Quadratic Reversible Centers of Genus One

Abstract

In this paper, using the method of Picard-Fuchs equation and Riccati equation, for a class of quadratic reversible centers of genus one, we research the upper bound of the number of zeros of Abelian integrals for the system $(r10)$ under arbitrary polynomial perturbations of degree $n$. Our main result is that the upper bound is $21n − 24 (n ≥ 3)$, and the upper bound depends linearly on $n$.