Linear relations among algebraic points on tensor powers of the Carlitz module

TL;DR

This paper investigates linear relations among algebraic points on tensor powers of the Carlitz module, employing Anderson dual t-motives and Frobenius difference equations to establish criteria for linear independence of Carlitz polylogarithms.

Contribution

It introduces new methods using Anderson dual t-motives and Frobenius equations to analyze linear independence of Carlitz polylogarithms at algebraic points.

Findings

Derived explicit sufficient conditions for linear independence of Carlitz polylogarithms.

Analyzed linear equations on tensor powers of the Carlitz module.

Applied the theory to both $ abla$-adic and $v$-adic settings.

Abstract

In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual $t$-motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some explicit sufficient conditions for the linear independence for Carlitz polylogarithms at algebraic points in both $\infty$-adic and $v$-adic settings.