On a characterization of polynomials among rational functions in non-archimedean dynamics

TL;DR

This paper characterizes polynomials among rational functions of degree greater than one over non-archimedean fields using dynamics and potential theory on the Berkovich projective line.

Contribution

It provides a new characterization of polynomials among rational functions in non-archimedean dynamics based on potential theory and Berkovich space analysis.

Findings

Polynomials can be distinguished from other rational functions via their dynamical properties.

The characterization leverages potential theory on the Berkovich projective line.

Results apply to algebraically closed fields with non-trivial non-archimedean absolute values.

Abstract

We study a question on characterizing polynomials among rational functions of degree $>1$ on the projective line over an algebraically closed field that is complete with respect to a non-trivial and non-archimedean absolute value, from the viewpoint of dynamics and potential theory on the Berkovich projective line.