一类四次Hamilton函数Abel积分的零点个数估计

曲靖师范学院学报 ›› 2021, Vol. 40 ›› Issue (3): 11-16.

一类四次Hamilton函数Abel积分的零点个数估计

王彦杰¹, 曹勃¹, 洪晓春²

1.宁波职业技术学院数学教研室,浙江宁波 315800;
2.云南财经大学统计与数学学院,云南昆明 650221

收稿日期:2021-04-08 出版日期:2021-05-26 发布日期:2021-07-13
作者简介:王彦杰,宁波职业技术学院数学教研室助教,主要从事常微分方程研究.
基金资助:
2020年宁波职业技术学院校级课题“四次Hamilton系统的极限环零点个数研究”(NZ21Q017).

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

WANG Yanjie¹, CAO Bo¹, HONG Xiaochun²

1. School of Statistics and Mathematics, Ningbo Polytechnic, Ningbo Zhejiang 315800;
2. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming Yunnan 650221, China

Received:2021-04-08 Published:2021-05-26 Online:2021-07-13

摘要/Abstract

摘要： 本文运用Picard-Fuchs方程法和Riccati方程法,研究了一类四次Hamilton函数H（x,y）=x²+y²+ax²y²+bx⁴+cy⁴的Abel积分零点个数问题,结论表明Abel积分在区间$(\frac{c}{a^{2}-4bc},0)$上,其零点个数不超过$3[\frac{n-1}{4}]+12[\frac{n-3}{4}]+22$.

关键词: Hamilton系统, Abel积分, Picard-Fuchs方程, Riccati方程

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

中图分类号:

O175

王彦杰, 曹勃, 洪晓春. 一类四次Hamilton函数Abel积分的零点个数估计[J]. 曲靖师范学院学报, 2021, 40(3): 11-16.

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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