On the Newton polytope of the Morse Discriminant

Within the framework of my PhD project, I have studied the tropical coun- terparts of A2 and 2A1 singularities of rational functions in one variable. One of my main results in this direction is the description of the Newton polytope of the Morse discriminant (the hypersurface containing all non-Morse Laurent polynomials) in the space of univariate polynomials with given support. Namely I have established a surjection of the set of all possible combinatorial types of so-called Morse tropical polynomials with the given support and the vertices of the Newton polytope of the Morse discriminant.