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 $\leq 4$, calculate all 71 possible values of these invariants for the case of degree four, and realize them by concrete Morse polynomials. Also we calculate the number of classes (up to isotopy and reflections in ${\mathbb R}^2$) of strictly Morse polynomials of degree four with the maximal possible number of real critical points.