The monotonicity of the ratio of hyperelliptic integrals
type: Item
instance of: scholarly article
title: The monotonicity of the ratio of hyperelliptic integrals
zbMATH Open document ID: 1316.34032
author: Na Wang
author: Dongmei Xiao
author: Jiang Yu
published in: Bulletin des Sciences Mathématiques
publication date: 2014-10-06T00:00:00Z
review text: The weakened Hilbert's 16th problem is to find the isolated zeros of the abelian integral \[ I(h)=\int_{\gamma(h)}^{}f(x,y)dx+g(x,y)dy, \] where \(f(x,y)\) and \(g(x,y)\) are polynomials. The curve \(\gamma(h)\) is a compact component of the level set \(H(x,y)=h\). The aim of this paper is study the monotonicity of the ratio \(P(h)=\frac{I_{1}(h)}{I_{0}(h)}\), where \[ I_{0}(h)=\oint_{\gamma(h)}^{} y dx, \,\,I_{1}(h)=\oint_{\gamma(h)}^{} xy dx. \] The systems under consideration are written in the form \[ \begin{aligned}\dot{x}&=-2y,\dot{y}=x(x-\alpha)(x-\beta)(x-\lambda)(x-1) \\ \dot{x}&=2y, \dot{y}=x(x-\alpha)(x-\beta)(x-\lambda)(x-1).\end{aligned} \] It is shown that \(P(h)\) is monotone for several cases of \(\alpha, \beta,\) and \( \lambda\). Some interesting conjectures are presented.
reviewed by: M. A. M. Alwash
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 14K20
zbMATH DE Number: 6351416
zbMATH Keywords: abelian integral
zbMATH Keywords: hyperbolic curve
zbMATH Keywords: weakened Hilbert's 16th problem
MaRDI profile type: Publication
full work available at URL: j.bulsci.2014.02.001
OpenAlex ID: W2046823797
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DOI: 10.1016/J.BULSCI.2014.02.001
description: scientific article; zbMATH DE number 6351416