Graduate Student Seminar - Andrew Herring (Math)

Dear Graduate Students,

You are all invited to the first talk for the summer term of the Graduate Student Seminar this Thursday 1:00-2:00PM via Zoom (meeting ID and password below). Our speaker this week, Andrew Herring, will talk about the Galois groups of certain special polynomials that appear in arithmetic dynamics.

Speaker: Andrew Herring

Title: Galois groups of Dynatomic Polynomials

Abstract: Arithmetic dynamics is a relatively young discipline which studies number theory questions associated to dynamical systems. Staring from a polynomial f with Q-coefficients, we might consider points P which are periodic under polynomial iteration: let f^n denote the n-fold composite of f with itself, and ask about those points for which f^n (P) = P. There’s a natural “generating function,” called the nth dynatomic polynomial, which has as its zero set the set of points P for which f^n (P) = P, but f^m (P) != P for m <n.

An active area of research in this field involves trying to compute Galois groups of dynatomic polynomials. As we will discuss, each such dynatomic Galois group naturally embeds in a “wreath product” of finite groups, and thus has a natural upper bound on its size. Some combination of heuristic arguments and experimental data support the idea that this dynatomic Galois group should (almost) always be as big as possible. Thus, arithmetic dynamicists are interested in being able to explicitly describe when this dynatomic Galois group is smaller than expected.

We hope to present a few simple examples of smaller-than-expected dynatomic Galois groups. Many examples will be computed in Sage in order to aid in the exposition. We will assume only basic familiarity with group theory and Galois theory.