Characterization of 2-ramified power series

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.