Some new results on Darboux integrable differential systems
type: Item
instance of: scholarly article
title: Some new results on Darboux integrable differential systems
zbMATH Open document ID: 1252.34001
author: Antoni Ferragut
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-01T00:00:00Z
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 6062829
zbMATH Keywords: planar polynomial differential system
zbMATH Keywords: Darboux first integral
zbMATH Keywords: inverse integrating factor
zbMATH Keywords: remarkable value
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.04.075
OpenAlex ID: W2074887129
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DOI: 10.1016/J.JMAA.2012.04.075
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review text: The author studies complex planar polynomial differential systems having a Darboux first integral. He defines remarkable values and remarkable curves associated with Darboux first integrals. Remarkable values and remarkable curves were first defined by Poincaré for rational first integrals; their importance was first shown in [\textit{J. Chavarriga, H. Giacomini, J. Giné} and \textit{J. Llibre}, J. Differ. Equations 194, No. 1, 116--139 (2003; Zbl 1043.34001)].NEWLINENEWLINEIn the present paper, these definitions are extended to Darboux first integrals, a result is proved that characterizes the existence of a polynomial inverse integrating factor by means of the number of critical remarkable values for these systems. The infinity is studied, through the so-called characteristic polynomial \(\mathcal{F}=x \tilde{Q}-y \tilde{P}\), where \(\tilde{P}\) and \(\tilde{Q}\) are the homogeneous parts of highest degree of \(P\) and \(Q\), the polynomials that define the differential system, and its relation with the inverse integrating factors of the system. The importance of the numerator of the exponential factor of the Darboux first integral for the construction of \(\mathcal{F}\) is shown.
description: scientific article; zbMATH DE number 6062829