About a conjecture on quadratic vector fields
type: Item
instance of: scholarly article
title: About a conjecture on quadratic vector fields
zbMATH Open document ID: 1011.12006
DOI: 10.1016/S0022-4049(00)00177-8
author: Jean Moulin Ollagnier
published in: Journal of Pure and Applied Algebra
publication date: 2003-05-27T00:00:00Z
reviewed by: Mykola Grygorenko
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 1698496
zbMATH Keywords: differentiation
zbMATH Keywords: Darboux element
zbMATH Keywords: partial differential field
Wikidata QID: Q123163471
MaRDI profile type: Publication
cites work: Q3336052
cites work: Q3476048
cites work: Integrals of quadratic ordinary differential equations in \(\mathbb R^3\): the Lotka-Volterra system
cites work: Equations de Pfaff algébriques
cites work: Polynomial first integrals of the Lotka-Volterra system
cites work: Rational integration of the Lotka-Volterra system
full work available at URL: s0022-4049(00)00177-8
OpenAlex ID: W2080927414
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review text: Let \(U\) be a partial differential field of characteristic 0 with differentiations \(\delta_{1},\dots ,\delta_{m}\) and \(R\) be a factorial differential subring of \(U\) with a field of quotients \(F\). Denote by \(C\) a field of constants of \(F\). For any \(r_{1},\dots ,r_{m} \in R, \delta=r_{1}\delta_{1}+\cdots +r_{m}\delta_{m}\) is a differentiation on \(R\). The differentiation \(\delta\) is called full if from \(\delta f = 0 \Rightarrow f\in C\), \(\forall f\in F\). An element \(f\in F\) is called a rational first integral of degree 0 (with respect to \(\delta\)) if \(f\notin C\) and \(\delta f=0\). An element \(t\in R\setminus C\) is called a Darboux element if \(\delta t=\lambda t\) for some \(\lambda\in R\). The answer to the question whether the given differentiation is full or not full is not trivial even if \(m=2\) [see \textit{A. V. Astrelin}, Mosc. Univ. Math. Bull. 48, No. 2, 24--27 (1993); translation from Vestn. Mosk. Univ., Ser. I 48, 27--31 (1993; Zbl 0792.12002)]. If \(R\) does not contain Darboux elements then it is differentially integrally closed (with respect to \(\delta\)) in \(F\) [see \textit{J. Brzezinski}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 10, 325--328 (1962; Zbl 0104.26301) and \textit{H. Nishimura}, Proc. Japan Acad. 40, 145--149 (1964; Zbl 0143.06101)]. The existence of the first integrals for \(\delta\) deprives \(R\) of the above property and, hence, causes the presence of Darboux elements. The opposite is incorrect, as the author shows in this paper. He considers the case where NEWLINE\[NEWLINER=C[x, y, z],\;\delta_{1}=\frac{\partial}{\partial x},\;\delta_{2}=\frac{\partial}{\partial y},\;\delta_{3}=\frac{\partial}{\partial z}NEWLINE\]NEWLINE and provides the whole family of full (Lotka-Volterra) differentiations of \(R\), which have Darboux elements.
description: scientific article; zbMATH DE number 1698496