Isotopy classification of Morse polynomials of degree 4 in ${\mathbb{R}}^2$

Abstract

We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb{R}}^2\to {\mathbb{R}}^1$, prove its completeness for polynomials of degrees $\le 4$, calculate all 71 possible values of these invariants for the case of degree 4, and realize them by concrete Morse polynomials. Also we count all 45460 classes of {\em strictly} Morse polynomials of degree four with the maximal possible number (nine) of real critical points. Keywords: real algebraic geometry, Morse function, Milnor fiber, Coxeter-Dynkin graph, vanishing cycle, topological invariant, surgery, Lyashko--Looijenga map.