EAPSI: Counting Pointed Dynamical Systems Over Finite Fields

Gunther Joseph, Brooklyn NY

Investigators

Abstract

This award supports research to advance understanding of fundamental constraints on dynamical systems over finite fields. Such systems play an important societal role; for example, they are central to the elliptic curve cryptography currently in use around the world. A dynamical system consists of an ambient space, along with a rule for how points in that space move over time. Dynamical systems constructed from finite fields are objects of fundamental mathematical interest, both theoretically and for practical applications to modern cryptography, which involves iterating a process that is hopefully hard to reverse. In studying a dynamical system, one often tries to understand the points that are the most well-behaved: fixed points, which do not move over time, and more generally periodic points, which are eventually brought back to their original locations after a finite amount of time. This project is a computational exploration of quadratic dynamical systems over finite fields, which are the simplest interesting examples, efficiently defined by a single quadratic polynomial. The researcher will examine patterns in the counts, over different finite fields, of how many systems having a given structure of periodic points there are. The project will be conducted at University of New South Wales in Sydney, Australia, under the mentorship of Professor John Roberts. This provides access to their uniquely strong group of researchers working in computational number theory along with important high-performance computating resources. These counts of pointed dynamical systems will be conducted over 1) finite fields whose sizes are growing powers of a fixed prime number, and 2) finite fields of prime size for different primes. The first approach can be used, thanks to the Lang-Weil estimates, to examine the irreducibility of reductions of dynatomical curves, important geometric objects which parameterized the dynamical systems under consideration. The second approach has been fruitful in recent years in uncovering unexpected geometric structure in sequences of moduli spaces, in particular in the case of the moduli space of curves of fixed genus with an increasing number of marked points. The researcher will also apply recently developed closed-point sieve methods to questions of how closely degree d polynomial dynamical systems can approximate random d-to-1 dynamical systems. This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the Australian Academy of Science.