Upper Bound of the Number of Zeros for Abelian Integrals in a Kind of Quadratic Reversible Centers of Genus One

Abstract

By using the methods of Picard-Fuchs equation and Riccati equation, we study the upper bound of the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under polynomial perturbations of degree $n.$ We obtain that the upper bound is $7[(n − 3)/2] + 5$ when $n ≥ 5, 8$ when $n = 4, 5$ when $n = 3, 4$ when $n = 2,$ and $0$ when $n = 1$ or $n = 0,$ which linearly depends on $n.$