Power series of finite order under substitution

Hybrid talk. For Zoom details please contact: Pieter Moree (moree@mpim...)

The problem we want to deal with is a very basic one: find formal power series over a finite field with p elements that are of finite order under substitution. Local class field theory proves existence, and explicit local class field theory provides an algorithm to compute successive coefficients.
We will focus instead on closed expressions, and, in particular, notions of complexity for these. The picture is rather bleak: Klopsch found representatives up to conjugation for all series of order p; and Jean and Chinburg-Symonds found an element of order 4 with smallest “breaks”; end of story.
We approach the problem by using automata, a very basic kind of Turing machine, based on Christol’s indentification of algebraic and automatic power series. This allows us to give a closed expression for, for example, elements of order 4 with higher breaks, an element of order 8, and a representation of the Vierergruppe by automata. Switching to automata also allows us to discuss how complex such a representation is, in terms of a notion of sparseness originally introduced in the theory of formal languages by Cobham, and recently translated to a field theoretic property by Albayrak and Bell. All previously know examples turn out to not be sparse, but we can find sparse representatives up to conjugation in certain infinite families.

The talk is based on joint work with Djurre Tijsma (Duesseldorf) and Jakub Byszewski (Krakow), and work of Mieke Wessel (Utrecht). Algorithms were developped in collaboration with Andrew Bridy (Yale) and Onno van Zomeren (Utrecht).