Graduate Student Seminar - Andrew Herring (Math)

Dear Graduate Students,

You are all invited to the second talk this summer 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 continue with Part 2 of last week's talk on the Galois groups of dynatomic polynomials.

Speaker: Andrew Herring

Title: Galois Groups of Dynatomic Polynomials II

Abstract: In last week's talk, we defined the dynatomic polynomial associated to a polynomial dynamical system, we discussed the orbit structure on the set of zeros of certain factors of the dynatomic polynomial, and began to look at Galois groups of such factors (henceforth called dynatomic Galois groups).  

This week, we continue where we left off: after recalling relevant information from last week, we will see that the dynatomic Galois group naturally embeds in a certain wreath product, and that we therefore have a natural upper bound on the size of the dynatomic Galois group.  It's a theorem due to Bousch (and later Morton) that the "generic" dynatomic polynomial for quadratic polynomial maps is irreducible.  We'll discuss how this generic irreducibility implies that the dynatomic Galois group is almost always "pretty big."  It is therefore an interesting question to ask about when dynatomic Galois groups are "small."  We'll meet a few "small" dynatomic Galois groups, and look at plenty of examples in Sage.