2016/11/03 by Kallal, Kenz, Kirkpatrick, Hudson
#11D88 #11S15 #11S82 (Primary) #20E18 (Secondary) #20E36 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1611.01077
Let K be a complete discrete valuation field with residue class field k, where both are of positive characteristic p. Then the group of wild automorphisms of K can be identified with the group under composition of formal power series over k with no constant term and X-coefficient 1. Under the hypothesis that p > b2, we compute the first nontrivial coefficient of the pth iterate of a power series over k of the form f = X + ∑i ≥ 1 aiXb+i. As a result, we obtain a necessary and sufficient condition for an automorphism to be ``b-ramified,'' having lower ramification numbers of the form in(f) = b(1 + ⋯ + pn). This is a vast generalization of Nordqvist's 2017 theorem on 2-ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in Cp and a 2015 result of Lindahl--Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl--Nordqvist's 2018 theorem bounding the norm of periodic points of 2-ramified power series.