Klassifizierung von polynomiellen Abbildungen mittels polyedrischer Geometrie
Zusammenfassung der Projektergebnisse
The theme of this project revolves around complex polynomial maps on the plane. That is, a map of the form f := (f1 , f2 ) : C2 → C2 , where each of f1 and f2 is a polynomial. Two such maps f, g : C2 → C2 are said to be topologically equivalent if there are homeomorphisms φ, ψ : C2 → C2 satisfying f = φ ◦ g ◦ ψ. It was proven in the 80’s that there are finitely-many topologically nonequivalent maps for any given degrees of the polynomials. Consequently, it is reasonable to ask about the number of such topolgical types of polynomial maps with fixed degrees. Due to the lack of dedicate tools, this problem turned out to be difficult for degrees higher than two. In this project, I have proposed to address the problem of developing tools for studying topological types of polynomial maps. The proposed approach would take advantage of the recent advancements of polyhedral geometry in contexts of algebraic geometry. Such results emanate from fields such as tropical geometry, toric geometry, theory of A-discriminants, and some results I have proven regarding the non-properness set of polynomial maps. The project was divided into two main objectives: the first one (Objective A) aimed at the use of toric geometry to develop a procedure for classifying topological types of polynomial maps. The second objective (Objective B) was aimed to describe a correspondence between tropical polynomial maps and complex polynomial maps. Such a description should relate topological type of a polynomial map to a polyhedral invariant in the tropical context. During the work on the project, it turned out that the time allocated for both objectives was too little. Consequently, the goals were achieved only partially. With Kemal Rose from MPI Leipzig, we are completing a work preparation (not yet submitted) that addresses Objective A. In this work, we achieve a description of topological types of any polynomial map satisfying some genericity condition with respect to its Newton polytope. In particular, we describe the singularities of its bifurcation set, and we use polyhedral methods to compute them and other invariants such as the genus of the critical locus. This is accompanied by a software implementation. Consequently, we obtain non-trivial and new lower bounds on the number of topological types of polynomial maps with a given degree. The results combined provide a correspondence theorem relating the bifurcation set of polynomial maps defined over a field of Puiseux series, to their tropical counterparts for tropical polynomial maps. This is done under a genericity assumption regarding the supports of the polynomials. Furthermore, I provide a method for computing the tropical discriminant and tropical non-properness set for polynomial maps. The results stated for the non-properness set are, in fact, stated for maps in higher dimension, which was beyond the scope of the project. In addition, I show how to apply this result to compute the dual fan of the Newton polytope of the bifurcation set of a general polynomial map with fixed supports. The achieved results show how one can use polyhedral tools to make non-trivial statements about the topology of polynomial maps if some genericity conditions are assumed. This suggests that the considered approach is still in its infancy and has pontential to make this connection deeper. Regarding Objective A, a new theory has to be learned to describe degenerate maps. As for Objective B, eventhough correspondence theorems have been proved, no tangible connection has been established between the topological types of maps and polyhedral types of tropical maps.
Projektbezogene Publikationen (Auswahl)
-
Computing the Non-properness Set of Real Polynomial Maps in the Plane. Vietnam Journal of Mathematics, 53(2), 245-275.
El Hilany, Boulos & Tsigaridas, Elias
-
Coupler curves of moving graphs and counting realizations of rigid graphs. Mathematics of Computation, 93(345), 459-504.
Grasegger, Georg; El Hilany, Boulos & Lubbes, Niels
