The right classification of univariate power series in positive characteristic

TL;DR

This paper provides a complete classification of univariate power series over algebraically closed fields of positive characteristic, revealing how support determines right determinacy and relating modality to the Milnor number.

Contribution

It introduces explicit normal forms for classification and establishes the relationship between right modality, Milnor number, and support in positive characteristic.

Findings

Classification is explicit via normal forms

Right determinacy depends on support

Modality equals the integer part of μ/p

Abstract

While the classification of univariate power series up to coordinate change is trivial in characteristic 0, this classification is very different in positive characteristic. In this note we give a complete classification of univariate power series $f\in K[[x]]$, where $K$ is an algebraically closed field of characteristic $p>0$ by explicit normal forms. We show that the right determinacy of $f$ is completely determined by its support. Moreover we prove that the right modality of $f$ is equal to the integer part of $\mu/p$, where $\mu$ is the Milnor number of $f$. As a consequence we prove in this case that the modality is equal to the proper modality, which is the dimension of the $\mu$-constant stratum in an algebraic representative of the semiuniversal deformation with trivial section.