Classifying polynomial maps by means of polyhedral geometry

• Applicant: Boulos El Hilany, Ph.D.
• Subject Area: Mathematics
• Term: from 2020 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 446593912

In Plato's allegory of the cave, people are trapped in a cave and can only see the shadows of movements from the real world outside the cave. These shadow images on the cave's wall constitute reality to them, as it's all they know - they cannot turn around to look outside the cave. Similarly, with topology of polynomial maps, we are restricted to looking at the bifurcation set - the shadows - to be able to make statements about the atypical fibres - the true objects casting the shadows. According to Plato, only the rare true philosopher will succeed in escaping from the cave into reality. For now, we will content ourselves with extrapolating from the shadows.This project concerns the study of polynomial maps from the complex plane to itself. That is, the coordinates of points in the image under those functions are polynomials in the coordinates of points in the source space. The topological type of such a map denotes the shape taken by its preimages' locus over the target plane.Distinguishing between topologically identical and topologically different maps is key for numerous applications in mathematics. For instance, each of the behaviours of iterated polynomial maps in dynamical systems, likelihood functions of an implicit statistical model, and solutions to optimization problems may vary considerably if their respective polynomial maps differ in topology.The lack of effective procedures to differentiate between those cases hinders any progress in the open topological classification problem of polynomial maps. In this project, I will develop a set of methods to fill this gap.In my opinion, an ideal base on which to build these approaches is an accurate characterization of the bifurcation set. That is the smallest set of points in the target space at which the preimage is a locally trivial fibration. Hence, the focal point of my project is a novel description of the bifurcation set. I have recently designed a combinatorial method to describe the part of the set producing atypical behaviour outside the complex plane. For the complementary part, I will adapt known techniques such as A-discriminants and toric geometry. By merging all these approaches, I will arrive at two mutually independent descriptions of the bifurcation set, designed for distinct applications:Firstly, a precise characterization suited for polynomials with simple structures. Secondly and essentially, four possible approaches to transform the problem into a classification of combinatorial types of tropical curves. This I will achieve by designing a correspondence theorem linking the topology of planar curves with the combinatorics of graphs.

• DFG Programme: WBP Position