Real Morse polynomials of degrees 5 and 6

Abstract

A real polynomial $p$ of degree $n$ is called a Morse polynomial if its derivative has $n-1$ pairwise differentreal roots and values of $p$ in these roots (critical values) are also pairwise different. The plot of such polynomial is called a "snake". By enumerating critical points and critical values in the increasing order we construct a permutation $a_1,\ldots,a_{n-1}$, where $a_i$ is the number of polynomial's value in $i$-th critical point. This permutation is called the \emph{passport} of the snake (polynomial). In this work for Morse polynomials of degrees 5 and 6 we describe the partition of the coefficient space into domains of constant passport.