Topological classification of Morse polynomials
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 \(4\).
scientific article; zbMATH DE number 5809092