Real Morse polynomials of degrees 5 and 6

TL;DR

This paper characterizes the structure of Morse polynomials of degrees 5 and 6 by partitioning their coefficient space into regions with identical passports, aiding understanding of their geometric and combinatorial properties.

Contribution

It provides a detailed description of the coefficient space partition for Morse polynomials of degrees 5 and 6 based on their passports, a novel classification approach.

Findings

Partition of coefficient space into domains of constant passport for degree 5 and 6.

Explicit description of passports for Morse polynomials of degrees 5 and 6.

Enhanced understanding of the geometric structure of Morse polynomials.

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.