Real Morse polynomials of degree $5$ and $6$

Yu. Yu. Kochetkov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2021) no. 4, pp. 99-112

Résumé

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