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| 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, Picard-Fuchs 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 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 $n\ge 5$, $8$ when $n=4$, $5$ when $n=3$, $4$ when $n=2$, and $0$ when $n=1$ or $n=0$, which linearly depends on $n$. |
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