On the number of invariant straight lines for polynomial differential systems
type: Item
instance of: scholarly article
title: On the number of invariant straight lines for polynomial differential systems
zbMATH Open document ID: 0930.34022
author: Joan C. Artés
author: Branko Grünbaum
author: Jaume Llibre
published in: Pacific Journal of Mathematics
publication date: 2000-02-09T00:00:00Z
review text: Let \(\alpha(n)\) be the maximum number of invariant straight lines possible in a polynomial differential system of degree \(n>1\) having finitely many invariant straight lines. By using a method in graph theory developed by the second author, the remarkable result: \(\alpha(5)= 14\) (together with many upper and lower bounds for \(\alpha(6),\alpha(7),\dots, \alpha(20)\)) is obtained. This result gives a negative answer to the old conjecture: \(\alpha(n)= 2n+1\), if \(n\) is even; \(\alpha(n)= 2n+2\), if \(n\) is odd. Moreover, the authors can prove that \(\alpha(n)\leq 3n-1\) for all \(n\). So, now they have the following results (the first three are known): \(\alpha(2)= 5= 3\cdot 2-1\), \(\alpha(3)= 8= 3\cdot 3-1\), \(\alpha(4)= 9< 3\cdot 4-1\), \(\alpha(5)= 14= 3\cdot 5-1\). There are interesting questions: Can we get a general formula for \(\alpha(n)\)? Is this question related to number theory?
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 1344237
zbMATH Keywords: polynomial differential system
zbMATH Keywords: invariant straight line
zbMATH Keywords: geometric straight line question
zbMATH Keywords: graph theory
reviewed by: Yanqian Ye
MaRDI profile type: Publication
full work available at URL: pjm.1998.184.207
OpenAlex ID: W1990139562
Wikidata QID: Q56442899
DOI: 10.2140/PJM.1998.184.207
description: scientific article; zbMATH DE number 1344237