# Residue fixed point index and wildly ramified power series

## TL;DR

This paper introduces a formula for the residue fixed point index of power series with fixed point multiplier 1 and applies it to analyze wildly ramified power series in positive characteristic, revealing their generic properties and bounds on periodic points.

## Contribution

It provides a closed-form expression for the residue fixed point index and characterizes power series with minimal ramification, establishing their genericity and bounds on periodic points.

## Key findings

- Derived a closed formula for the residue fixed point index.
- Characterized power series with minimal ramification numbers.
- Established genericity and bounds for periodic points in convergent series.

## Abstract

In this paper, we study power series having a fixed point of multiplier 1. First, we give a closed formula for the residue fixed point index, in terms of the first coefficients of the power series. Then, we use this formula to study wildly ramified power series in positive characteristic. Among power series having a multiple fixed point of small multiplicity, we characterize those having the smallest possible lower ramification numbers in terms of the residue fixed point index. Furthermore, we show that these power series form a generic set, and, in the case of convergent power series, we also give an optimal lower bound for the distance to other periodic points.