Computation of the Galois groups occuring in M. Papanikolas's study of Carlitz logarithms

TL;DR

This paper provides a method to compute the unipotent radical of Galois groups in a positive characteristic Tannakian setting and applies it to prove algebraic independence of Carlitz logarithms.

Contribution

It introduces a theorem for computing Galois group radicals and offers criteria for algebraicity of solutions, with applications to $t$-motives and Carlitz logarithms.

Findings

Computed unipotent radicals of Galois groups in positive characteristic.

Established criteria for algebraicity of solutions based on linear dependence.

Provided an alternative proof of algebraic independence of Carlitz logarithms.

Abstract

In this note, we state a theorem of compution of the unipotent radical of the Galois group of an object $U$ of a tannakian category defined over a field of positive characteristic, extension of the unit object by a semi-simple one. We then give a criteria of algebricity of solutions of the objects in terms of linear dependences in groups of extension. We apply these results to the tanakian framework of $t$-motives developped by M. Papanikolas; in particular, we give an alternative proof of the algebraic independence of Carlitz logarithms.