Characterization of 2-ramified power series

TL;DR

This paper characterizes power series tangent to the identity over fields of positive characteristic with specific ramification numbers, providing explicit formulas and insights into their iterates.

Contribution

It offers a complete characterization of such power series with particular ramification numbers and computes their first significant terms at the p-th iterate.

Findings

Characterization of power series with ramification numbers of the form 2(1+p+...+p^n)

Explicit computation of the first significant terms at the p-th iterate

Insight into the structure of ramified power series in positive characteristic fields

Abstract

In this paper we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristics $p$. Let $g$ be such a series, then $g$ has a fixed point at the origin and the corresponding lower ramification numbers of $g$ are then, up to a constant, the degree of the first non-linear term of $p$-power iterates of $g$. The result is a complete characterization of power series $g$ having ramification numbers of the form ${2(1+p+\dots+p^n)}$. Furthermore, in proving said characterization we explicitly compute the first significant terms of $g$ at its $p$th iterate.