An improvement to Darboux integrability theorem for systems having a center
type: Item
instance of: scholarly article
title: An improvement to Darboux integrability theorem for systems having a center
zbMATH Open document ID: 0978.34022
DOI: 10.1016/S0893-9659(99)00127-5
published in: Applied Mathematics Letters
publication date: 2000-04-09T00:00:00Z
review text: Let \(\dot x= P(x,y)\), \(\dot y= Q(x, y)\) be a plane polynomial differential system \((S)\) of degree \(m\). Darboux proved that, if \((S)\) has \({m(m+1)\over 2}\) algebraic solutions, then it has a first integral or an integrating factor of the type \(f^{\lambda_1}_1\cdots f^{\lambda_q}_q\), \(f_i\in \mathbb{C}[x, y]\), \(\lambda_i\in \mathbb{C}\). Such functions are called Darboux first integrals, or, respectively, Darboux integrating factors. During the last decade, Darboux' result was extended in several ways. Here, the following theorem is proved: Let \((S)\) have a center. If \((S)\) has \({m(m+1)\over 2}-[{m+1\over 2}]\) algebraic solutions or exponential factors, then \((S)\) has a Darboux integrating factor.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C08
zbMATH DE Number: 1428534
zbMATH Keywords: Darboux first integrals
zbMATH Keywords: algebraic solutions
zbMATH Keywords: exponential factors
author: Javier Chavarriga
author: Hector Giacomini
author: Jaume Giné
reviewed by: Marco Sabatini
MaRDI profile type: Publication
cites work: Algebraic Particular Integrals, Integrability and the Problem of the Center
cites work: Equations de Pfaff algébriques
cites work: Elementary First Integrals of Differential Equations
cites work: Algebraic solutions for polynomial systems with emphasis in the quadratic case
cites work: Invariant algebraic curves and conditions for a centre
cites work: Q4948951
cites work: Algebraic invariant curves and the integrability of polynomial systems
cites work: Partial integrals and the first focal value in the problem of centre
cites work: Q4350318
full work available at URL: s0893-9659(99)00127-5
OpenAlex ID: W1968483157
description: scientific article; zbMATH DE number 1428534