Real Morse Polynomials of Degree 5 and 6

Abstract

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 a1, . . . , an−1, where ai is the number of the polynomial’s value at the ith 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.