Twin quadratic polynomial vector fields in the space \(\mathbb{C}^3\)
type: Item
instance of: scholarly article
title: Twin quadratic polynomial vector fields in the space \(\mathbb{C}^3\)
zbMATH Open document ID: 1548.34024
author: Jaume Llibre
author: Claudia Valls
published in: Dynamics of Continuous, Discrete \& Impulsive Systems. Series A. Mathematical Analysis
publication date: 2024-07-03T00:00:00Z
review text: {\em Twin vector fields} are two different vector fields that agree on position and spectra at all their singularities. In several previous papers this notion has been studied for planar quadratic vector fields and for some vector fields in dimension \(3\). In the present paper the authors study this notion for quadratic vector fields in \(\mathbb{C}^3\) having exactly \(8\) singular points. By Bézout's Theorem, eight is the maximum number of isolated singularities that a quadratic vector field in \(\mathbb{C}^3\) can have and, thus, all the singularities are non-degenerated, that is, their associated eigenvalues are all different from zero. \N\NIn this family of vector fields, the authors fix as a hypothesis the position and spectra of \(M\) singular points and also some conditions on their associated eigenvalues. They study the cases when \(M=7\), \(M=6\) and \(M=5\). They also provide a theorem establishing the existence of twin vector fields inside the considered family. One of the main tools in the proof of their results is Euler-Jacobi formula.
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
zbMATH DE Number: 7875573
zbMATH Keywords: quadratic polynomial vector fields
zbMATH Keywords: singularities
zbMATH Keywords: spectra
zbMATH Keywords: twin vector fields
MaRDI profile type: Publication
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cites work: Twin vector fields and independence of spectra for quadratic vector fields
description: scientific article; zbMATH DE number 7875573