On the Abelian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree

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In this study, we determine the number of zeros for Abelian integrals in four cases of quadratic reversible centers of genus one. Based on the results of Li et al., we prove that the upper bound of the associated number of zeros for Abelian integrals with orbits formed by quartics, quintics, sextics, and eight-degree curves under polynomial perturbations of arbitrary degree n depends linearly on n.