The Newton Polytope of the Morse Discriminant of a Univariate Polynomial

TL;DR

This paper computes the Newton polytope of the Morse discriminant for univariate polynomials, linking combinatorial types of tropical polynomials to the vertices of this polytope.

Contribution

It establishes a surjection between Morse tropical polynomial types and the vertices of the Newton polytope of the Morse discriminant.

Findings

Identifies the vertices of the Newton polytope $\

Establishes a correspondence with combinatorial types of Morse tropical polynomials

Provides a geometric understanding of the Morse discriminant in polynomial space.

Abstract

In this paper we compute the Newton polytope $\mathcal M_A$ of the Morse discriminant in the space of univariate polynomials with the given support set $A.$ Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of $\mathcal M_A.$