On the invariant straight lines of polynomial differential systems
type: Item
instance of: scholarly article
title: On the invariant straight lines of polynomial differential systems
zbMATH Open document ID: 0903.34027
author: Joan C. Artés
author: Branko Grünbaum
author: Jaume Llibre
published in: Differential Equations and Dynamical Systems
publication date: 1998-08-10T00:00:00Z
review text: There is an interesting but difficult problem: to find the maximum number \(\alpha (n)\) of invariant straight lines in the set of all polynomial differential systems of degree \(n>1\) having finitely many invariant straight lines. Many authors proved independently that \(\alpha (2)=5\), \(\alpha (3)=8\), \(\alpha (4)=9\). Later, in 1995, H. Zoladek proved that \(\alpha (n)\leq 3n-1\). In this paper the authors give chiefly the following results: 1. \(2n+1\leq \alpha (n)\leq 3n-1\) for \(n\) even, 2. \(2n+2 \leq\alpha (n)\leq 3n-1\) for \(n\) odd, 3. \(\alpha (5)=14\). It is interesting to note that for \(n=2,3\) and 5, the authors have \(\alpha (n)=3n-1\) but for \(n=4\), \(\alpha (4)=9 <3\times 4-1\). Therefore, one may ask: How this problem relates to number theory?
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 1158166
zbMATH Keywords: polynomial differential systems
zbMATH Keywords: invariant straight lines
reviewed by: Yanqian Ye
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1158166