The set of all non-smooth hypersurfaces given by polynomials with the fixed support set A was described by Gelfand, Kapranov and Zelevinsky around 30 years ago. It gave rise to many interesting questions that have been addressed by various authors and quite a lot of them have only recently been resolved.

In this thesis we study a seemingly similar object – the Newton polytope of the Morse discriminant, the hypersurface, which is the closure of all non-Morse Laurent polynomials in the space of polynomials with given support A. Namely, we have made the first steps in developing a theory for this object, similar to the theory of Gelfand–Kapranov–Zelevinsky for the A-discriminant. The main result of the thesis is an explicit formula for the support function of the Newton polytope of the Morse discriminant for an arbitrary dimension 1 set A: Chapter 3 is devoted to the statement and the proof of this result.

Despite the similarity of the statements, the problems of describing the Newton polytopes of the A-discriminant and the Morse discriminant require different methods. The approach that we present in the thesis for the case of univariate polynomials has a very important advantage – it is in principle applicable to multivariate polynomials with given support.

We reduce computing the support function of the Newton polytope of the Morse discriminant to the problem of enumerating the singularities of a plane projection of a complete intersection curve given by a tuple of polynomials with fixed support. This problem is of its own interest, therefore, we address it in full generality in Chapter 2.