Darboux theory of integrability for polynomial vector fields in taking into account the multiplicity at infinity
type: Item
instance of: scholarly article
title: Darboux theory of integrability for polynomial vector fields in taking into account the multiplicity at infinity
zbMATH Open document ID: 1198.34054
author: Jaume Llibre
author: Xiang Zhang
published in: Bulletin des Sciences Mathématiques
publication date: 2009-12-01T00:00:00Z
review text: Darboux's theory of integrability provides a relation between the existence of invariant algebraic hypersurfaces of polynomial vector fields defined in \({\mathbb R}^n\) or \({\mathbb C}^n\), \(n\geq2,\) and the existence of first integrals. In a recent paper [J. Differ. Equations 246, No.~2, 541--551 (2009; Zbl 1163.34003)] the same authors improved the classical results taking into account the algebraic multiplicity of invariant algebraic hypersurfaces. In this paper, they extend these results by introducing the algebraic multiplicity of the hyperplane at infinity. Their main result is: Assume that a polynomial vector field \(X\) of degree \(d>0\) in \({\mathbb R}^n\) has \(p\) irreducible invariant hypersurfaces, \(f_1=0,f_2=0,\dots,f_p=0\), and an invariant hyperplane at infinity. Then: (i) If one of these \(p+1\) hypersurfaces has no defined algebraic multiplicity, then \(X\) has a rational first integral; (ii) otherwise, let \(q_i\) be the algebraic multiplicity of \(f_i\) and let \(k\) be the algebraic multiplicity of the hyperplane at infinity. If moreover, when one of these algebraic multiplicities is larger than 1, \(X\) restricted to the corresponding hypersurface has no rational first integral, then: {\parindent7mm \begin{itemize}\item[(a)] If \(\sum_{i=1}^p q_i+k\geq N+2\), then \(X\) has a real Darboux first integral, where \(N={n+d-1 \choose n}\), \item[(b)] If \(\sum_{i=1}^p q_i+k\geq N+n+1\), then \(X\) has a real rational first integral. \end{itemize}} In the paper, the algebraic multiplicity of an algebraic hypersurface \(f=0\) of degree \(m\) is introduced as the multiplicity \(f\) as a factor of the \(m\)-th extactic polynomial associated to \(X\). If this polynomial is identically zero, then it is said that \(f\) has no defined algebraic multiplicity. It is also shown, through some examples, that the hypothesis on the non-existence of a rational first integral, when some hypersuface has algebraic multiplicity greater than 1, is necessary in the…
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 5641311
zbMATH Keywords: polynomial vector field in projective space
zbMATH Keywords: Darboux integrability
zbMATH Keywords: invariant algebraic hypersurface
reviewed by: Armengol Gasull
MaRDI profile type: Publication
full work available at URL: j.bulsci.2009.06.002
OpenAlex ID: W2009453463
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DOI: 10.1016/J.BULSCI.2009.06.002
description: scientific article; zbMATH DE number 5641311