Topological classification of Morse polynomials
type: Item
instance of: scholarly article
title: Topological classification of Morse polynomials
zbMATH Open document ID: 1214.57026
DOI: 10.1134/S0081543810010049
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2010-11-01T00:00:00Z
review text: At present, there is no known general topological classification of polynomials of low degree in two real variables, despite of the obvious connections to Hilbert's 16-th problem. The classification problem for polynomials in two variables is closely related to the problem of topological classification of smooth functions \(f: S^2\to \mathbb R\) on the sphere. In the paper under review the author discusses the topological classification of polynomials of degree 4 in two variables having non-degenerate critical points and distinct critical values. The author proves that out of 17746 topological types of smooth Morse functions \(f: S^2\to \mathbb R\) with the same number of critical points, at most 426 types can be realised by polynomials of degree \(4\).
Mathematics Subject Classification ID: 57R35
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57R70
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 5809092
zbMATH Keywords: Morse function
zbMATH Keywords: topological classifications
zbMATH Keywords: polynomials of degree 4
author: Vładimir I. Arnold
reviewed by: Michael Farber
MaRDI profile type: Publication
cites work: Smooth functions statistics
cites work: Topological classification of Morse functions and generalisations of Hilbert's 16-th problem
cites work: Q3123459
full work available at URL: s0081543810010049
OpenAlex ID: W2099912383
description: scientific article; zbMATH DE number 5809092