Estimation of the Number of Zeros for Abel Integrals of Quadratic Hamiltonian Functions

Abstract

Abstract: In this paper, we use the Picard-Fuchs equation method and the Riccati equation method to study a class of four-time Hamiltonian function H（x,y）=x²+y²+ax²y²+bx⁴+cy⁴, the zero number of Abel integrals of such functions. The conclusion shows that the Abel integral is in the interval $(\frac{c}{a^{2}-4bc},0)$, the number of zeros does not exceed $3[\frac{n-1}{4}]+12[\frac{n-3}{4}]+22$.

Key words: Hamilton system, Abel integral, Picard-Fuchs equation, Riccati equation

CLC Number:

O175

WANG Yanjie, CAO Bo, HONG Xiaochun. Estimation of the Number of Zeros for Abel Integrals of Quadratic Hamiltonian Functions[J]. Journal of Qujing Normal University, 2021, 40(3): 11-16.

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