Number of invariant straight lines for homogeneous polynomial vector fields of arbitrary degree and dimension
type: Item
instance of: scholarly article
title: Number of invariant straight lines for homogeneous polynomial vector fields of arbitrary degree and dimension
zbMATH Open document ID: 1183.34062
author: Jaume Llibre
author: Violetta Pilyugina
published in: Journal of Dynamics and Differential Equations
publication date: 2009-10-14T00:00:00Z
review text: The authors consider in \(\mathbb{R}^d\) (respectively \(\mathbb{C}^d\)) the real (respectively complex) polynomial differential system of degree \(m\): \[ \begin{aligned} \dot x_1 & =P_1(x_1,\dots,x_d), \\ &\quad \cdots\cdots \\ \dot x_d & =P_d(x_1,\dots,x_d),\end{aligned} \] where \(m\geq1\) is the maximum of the degrees of the polynomials \(P_i\) for \(i=1,\dots,d\). In the case that all polynomials \(P_i\) are homogeneous of degree \(m\), they determine the number of invariant straight lines through the origin based on Bézout's theorem in algebraic geometry, when this number is finite. They also show that this number provides an upper bound for the number of infinite singular points of the polynomial differential systems of degree \(m\) in \(\mathbb{R}^d\), and this upper bound is reached if all the invariant straight lines through the origin are real.
reviewed by: Feng Xie
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5615000
zbMATH Keywords: homogeneous polynomial vector field
zbMATH Keywords: polynomial vector field
zbMATH Keywords: invariant straight line
zbMATH Keywords: infinite singular point
MaRDI profile type: Publication
full work available at URL: s10884-009-9141-x
OpenAlex ID: W2062732613
cites work: Bounded Polynomial Vector Fields
cites work: Algebraic and topological classification of the homogeneous cubic vector fields in the plane
DOI: 10.1007/S10884-009-9141-X
description: scientific article; zbMATH DE number 5615000