A LINEAR ESTIMATION TO THE NUMBER OF ZEROS FOR ABELIAN INTEGRALS IN A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE

Lijun Hong; Xiaochun Hong; Junliang Lu

doi:10.11948/20190247

2020 Volume 10 Issue 4

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Lijun Hong, Xiaochun Hong, Junliang Lu. A LINEAR ESTIMATION TO THE NUMBER OF ZEROS FOR ABELIAN INTEGRALS IN A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1534-1544. doi: 10.11948/20190247

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Lijun Hong, Xiaochun Hong, Junliang Lu. A LINEAR ESTIMATION TO THE NUMBER OF ZEROS FOR ABELIAN INTEGRALS IN A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1534-1544. doi: 10.11948/20190247

A LINEAR ESTIMATION TO THE NUMBER OF ZEROS FOR ABELIAN INTEGRALS IN A KIND OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE

School of Statistics and Mathematics, Yunnan University of Finance and Economics, 650221 Kunming Yunnan, China

Corresponding author: Email address: xchong@ynufe.edu.cn(X. Hong)
Fund Project: This work was supported by the National Natural Science Foundation of China (No. 11761075)

Abstract

Abstract

In this paper, using the method of Picard-Fuchs equation and Riccati equation, we consider the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under arbitrary polynomial perturbations of degree $n$, and obtain that the upper bound of the number is $2\left[{(n+1)}/{2}\right]+$ $\left[{n}/{2}\right]+2$ ($n\geq 1$), which linearly depends on $n$.
- Abelian integral /
- quadratic reversible center /
- weakened Hilbert's 16th problem /
- limit cycle
MSC: 34C07, 34C08, 37G15

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