Darboux integrability for polynomial vector fields invariant under action of finite group
type: Item
instance of: scholarly article
title: Darboux integrability for polynomial vector fields invariant under action of finite group
zbMATH Open document ID: 1266.34003
published in: Qualitative Theory of Dynamical Systems
publication date: 2013-02-05T00:00:00Z
full work available at URL: http://www.locus.ufv.br/handle/123456789/21337
review text: It is clear that a polynomial vector field of degree \(d\) in \(K^n\), with \(K \, = \, \mathbb{R}\) or \(K \, = \, \mathbb{C}\), admits a rational first integral if and only if it possesses an infinite number of invariant irreducible algebraic hypersurfaces. [\textit{J. P. Jouanolou}, Equations de Pfaff algébriques. Lecture Notes in Mathematics. 708. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0477.58002)] showed that, if the number of invariant irreducible hypersurfaces is finite, then it is bounded by \[ \left( \begin{matrix} d+n-1 \\ n \end{matrix}\right) + n. \] The present paper deals with polynomial vector fields which are indeed invariant under the action of a finite group \(G\). Denote by \(K_r[z]^G\) the space of \(G\)-invariant polynomials of degree \(r\). The main results in the paper is the following statement. If a \(G\)-invariant polynomial vector field admits \[ \dim_{K} \left( K_{d-1}[z]^G\right) + n \] invariant irreducible algebraic \(G\)-hypersurfaces, then it admits a \(G\)-rational first integral.
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 6134571
zbMATH Keywords: Darboux integrability
zbMATH Keywords: rational first integral
zbMATH Keywords: finite group action
author: Maurício jun. Corrêa
MaRDI profile type: Publication
OpenAlex ID: W2094209472
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DOI: 10.1007/S12346-011-0065-6
description: scientific article; zbMATH DE number 6134571