Wildly ramified power series with large multiplicity

Jonas Nordqvist

doi:10.1016/j.jnt.2021.01.019

Abstract

In this paper we consider wildly ramified power series, 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 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.

Highlights

In this paper we are interested in formal power series having a parabolic fixed point at the origin, such that ′(0) = 1
We say that two elements, ∈ (1+ [[ ]]) are formally conjugated if there exists a formal power series h of the form h(0) = 0, h′(0) ≠ 0 such that = h−1◦ ◦h
The multiplicity of is denoted by mult( ), and we put mult( ) ∶= ∞. Another property of that is invariant under conjugation is the residue fixed point index which is defined as the coefficient of 1 in the Laurent expansion about 0 of and denoted by index( ), see e.g. [Mil06, §12] for a description of the residue fixed point index.i Let ∶= Ĉ = C ∪{∞}, for a rational map ∈ ( ), not the identity, of the form (0) = 0 and ′(0) = 1, classification under formal conjugation is described in terms of the multiplicity and the residue fixed point index

Summary

INTRODUCTION

The multiplicity of (at the origin) is denoted by mult( ), and we put mult( ) ∶= ∞ Another property of that is invariant under conjugation is the residue fixed point index which is defined as the coefficient of 1 in the Laurent expansion about 0 of. Every wildly ramified power series has an associated sequence of integers called the lower ramification numbers. It encodes the multiplicity of the origin for the iterates of the power series. Ramified power series and its lower ramification numbers have been studied in many papers, see e.g., [Sen[69], Kea[92], LS98, Win04] for iWe further note that the residue fixed point index may be defined in a similar manner for any fixed point of .

MAIN RESULTS

APPLICATIONS

THE SECOND RESIDUE FIXED POINT INDEX AND ITS GENERALIZATIONS

LOWER RAMIFICATION NUMBERS FOR POWER SERIES WITH LARGE MULTIPLICITY