Wildly ramified power series with large multiplicity

TL;DR

This paper introduces a new invariant called the second residue fixed point index for wildly ramified power series in positive characteristic, providing a formula and characterizing those with minimal multiplicity under iteration.

Contribution

It defines the second residue fixed point index, derives a closed formula, and characterizes power series with large multiplicity that maintain minimal multiplicity upon iteration.

Findings

Introduced the second residue fixed point index invariant.

Provided a closed formula for the invariant.

Characterized power series with large multiplicity and minimal iterative multiplicity.

Abstract

In this paper we consider wildly ramified power series, \emph{i.e.}, power series defined over a field of positive characteristic, fixing the origin, where it is tangent to the identity. In this setting we introduce a new invariant under change of coordinates called the \emph{second residue fixed point index}, and provide a closed formula for it. As the name suggests this invariant is closely related to the residue fixed point index, and they coincide in the case that the power series have small multiplicity. Finally, we characterize power series with large multiplicity having the smallest possible multiplicity at the origin under iteration, in terms of this new invariant.