Polynomial differential systems having a given Darbouxian first integral
type: Item
instance of: scholarly article
title: Polynomial differential systems having a given Darbouxian first integral
zbMATH Open document ID: 1076.34002
author: Jaume Llibre
author: Chara Pantazi
published in: Bulletin des Sciences Mathématiques
publication date: 2005-02-09T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/1212
review text: A function \(H(x,y)=f_1^{\lambda_1}\cdots f_p^{\lambda_p} \exp(g/(f_1^{n_1}\cdots f_p^{n_p}) ),\) where \(f_1,\ldots,f_p\) are irreducible polynomials in \({\mathbb C}[x,y],\) \(\lambda_1,\ldots,\lambda_p\in\mathbb{C},\) \(n_1,\ldots,n_p\in{\mathbb N}\cup\{0\}\) and the polynomial \(g\in {\mathbb C}[x,y]\) is coprime with each \(f_i,\) whenever \(n_i\neq0,\) is called a Darbouxian function. The Darbouxian theory of integrability allows one to determine when a polynomial vector field has a first integral or an integrating factor of Darbouxian type. This paper mainly deals with the inverse problem, namely to characterize the polynomial vector fields \(X=(P,Q)\) in \({\mathbb C}^2\) which admit a first integral of Darbouxian type. The main result is an explicit expression of \(X\) which depends on the fact that \(\max\left(\deg g, \deg(f_1^{n_1}\cdots f_p^{n_p})\right)+\sum_{i=1}^p\deg{f_i}\) is greater or equal \(\deg(X)+1.\) One of the ingredients of the Darboux theory of integrability is the existence of a suitable number of invariant algebraic curves. The paper also deals with the search of Darbouxian integrating factors and gives a result which allows one to ensure the existence of such factors taking into account, among other things, not only the number of invariant algebraic curves, but also their degrees.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 2132853
zbMATH Keywords: polynomial differential system
zbMATH Keywords: first integral
zbMATH Keywords: Darbouxian function
zbMATH Keywords: integrating factor
MaRDI profile type: Publication
OpenAlex ID: W2126797173
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DOI: 10.1016/J.BULSCI.2004.04.001
description: scientific article; zbMATH DE number 2132853