Volume 8, Number 6, 2018, Pages 19591970 DOI：10.11948/2018.1959 
Upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one 
Xiaochun Hong,Junliang Lu,Yanjie Wang 
Keywords:Abelian integral, quadratic reversible center, weakened Hilbert''s 16th problem. 
Abstract: 
In this paper, by using the method of PicardFuchs 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 $3n3$ ($n\geq 2$) and $18\left[\frac{n}{2}\right]+3\left[\frac{n1}{2}\right]$ ($n\geq 4$) respectively, both of the two upper bounds linearly depend on $n$. 
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