Mahler's and Koksma's classifications in fields of power series

TL;DR

This paper explores the analogues of Mahler's and Koksma's classifications for power series over finite fields, establishing their equivalence and addressing a question posed by Ooto.

Contribution

It demonstrates that Mahler's and Koksma's classifications coincide in the context of power series over finite fields, providing a significant theoretical insight.

Findings

Both classifications are shown to coincide.

Addresses and resolves a question by Ooto.

Advances understanding of power series classifications in finite fields.

Abstract

Let $q$ a prime power and ${\mathbb F}_q$ the finite field of $q$ elements. We study the analogues of Mahler's and Koksma's classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.