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

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KeyWord:Abelian integral, quadratic reversible center, weakened Hilbert's 16th problem, PicardFuchs equation, Riccati equation 
Author Name  Affiliation  Qiuli Yu  School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China  Houmei He  School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China  Yuangen Zhan  Department of Information Engineering, Jingdezhen Ceramic University, Jingdezhen, Jiangxi 333403, China  Xiaochun Hong  School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China 

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Abstract: 
By using the methods of PicardFuchs 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 $7[(n3)/2]+5$ when $n\ge 5$, $8$ when $n=4$, $5$ when $n=3$, $4$ when $n=2$, and $0$ when $n=1$ or $n=0$, which linearly depends on $n$. 


