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

doi:10.11948/2018.1959

2018 Volume 8 Issue 6

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Xiaochun Hong, Junliang Lu, Yanjie Wang. UPPER BOUNDS FOR THE ASSOCIATED NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR TWO CLASSES OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1959-1970. doi: 10.11948/2018.1959

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Xiaochun Hong, Junliang Lu, Yanjie Wang. UPPER BOUNDS FOR THE ASSOCIATED NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR TWO CLASSES OF QUADRATIC REVERSIBLE CENTERS OF GENUS ONE[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1959-1970. 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

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

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Abstract

Abstract

In this paper, by using the method of Picard-Fuchs 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 3n -3 (n ≥ 2) and 18[n/2]n/2 + 3[(n-1)/1] (n ≥ 4) respectively, both of the two upper bounds linearly depend on n.
- Abelian integral /
- quadratic reversible center /
- weakened Hilbert's 16th problem
MSC: 34C07;34C08;37G15

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