# Constructing Separable Arnold Snakes of Morse Polynomials

## TL;DR

This paper presents a constructive method to realize any separable alternating permutation as the Arnold snake of a Morse polynomial, linking polynomial critical points with permutation patterns.

## Contribution

It provides a new constructive proof for the existence of Morse polynomials with prescribed critical value arrangements, specifically for separable Arnold snakes.

## Key findings

- Any separable alternating permutation can be realized as an Arnold snake of a Morse polynomial.
- The proof offers a constructive approach to associate polynomial critical points with permutation structures.
- The work bridges permutation patterns with the topology of polynomial graphs.

## Abstract

We give a new and constructive proof of the existence of a special class of univariate polynomials whose graphs have preassigned shapes. By definition, all the critical points of a Morse polynomial function are real and distinct and all its critical values are distinct. Thus we can associate to it an alternating permutation: the so-called Arnold snake, given by the relative positions of its critical values. We realise any separable alternating permutation as the Arnold snake of a Morse polynomial.