On the Newton Polytope of the Morse Discriminant

Diplôme Thèse On the Newton Polytope of the Morse Discriminant eng 2022-08-18 author Voorhaar Arina 853016 director Alexeev Anton 134019 79 10.13097/archive-ouverte/unige:164995 urn:nbn:ch:unige-1649957 Sc. 5673 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. Newton polytopes Elimination theory Tropical geometry Singularities Dewey 510 25 Section de mathématiques 11 false 168a233b-3b08-4094-a238-e5a845634922 Thesis Public thesis.pdf 899165 application/pdf CC BY 302d8135-8f93-4ab9-ad4c-2f5adcc5ff88 Imprimatur Public Imprimatur.pdf 121569 application/pdf openAccess Faculté des sciences / Section de mathématiques

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Voorhaar, Arina

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