On July 1, 2018 my stand-alone project P 30205-NBL, "Arithmetic Dynamical Systems, Polynomials, and Polytopes", of the Austrian Science Fund (FWF) began. An overview of activities and results of the project are presented here and will be updated regularly.
Overview
One of the main goals of this project is to introduce and study a new kind of arithmetic dynamical system which generalizes two seemingly unrelated notions: p-adic systems and permutation polynomials. The question for the ultimate behavior of p-adic systems ranges from trivial (base p expansions) to incredibly hard (3n+1 problem). Putting many known p-adic systems and permutation polynomials under a common framework might give a better understanding of the true nature of this great discrepancy. Furthermore, it might allow notions and methods of one discipline to be translated to the other admitting new points of views in both areas.
Another main focus will be the study of the Schur-Cohn region and related objects from a geometrical perspective. The Schur-Cohn region is defined by the set of coefficient vectors of contractive polynomials and is intimately related to dynamical properties of another kind of arithmetic dynamical system: shift radix systems. The volumes of the parts of a certain subdivision of the Schur-Cohn region hold many surprises and indicate a deep geometrical truth to be revealed.
A recent general result of Michael Kerber, Robert Tichy and myself on constrained triangulations of convex polytopes allowed the exact computation of an arithmetic constant from the theory of S-unit equations. Several open problems such as the question for the volume of the Birkhoff polytopes appear to be of a similar nature and might allow to be treated by this newly developed method. Exploring its applicability and possibly solving related questions will be a further aim of this project.
Activities
2018 (July - December)
- Project website created
- Article "The finiteness property for shift radix systems with general parameters" submitted
- Article "Constrained triangulations, volumes of polytopes, and unit equations" submitted
- Seminar on homotopy type theory
2019
- Article "The finiteness property for shift radix systems with general parameters" accepted
- Article "An introduction to p-adic systems - A new kind of number systems inspired by the Collatz 3n+1 conjecture" submitted
- Seminar on homotopy type theory