On algebraic independence of Taylor coefficients of certain Anderson-Thakur series

TL;DR

This paper investigates the algebraic independence of Taylor coefficients of Anderson-Thakur series, using t-motives and Papanikolas' theory to establish new independence results related to multiple zeta values in positive characteristic.

Contribution

It introduces a novel approach to proving algebraic independence of Taylor coefficients of Anderson-Thakur series via t-motives and Galois group dimension analysis.

Findings

Determined the dimension of t-motivic Galois groups under certain conditions.

Established algebraic independence of Taylor coefficients of Anderson-Thakur series.

Connected hyperderivatives of Anderson-Thakur series to multiple zeta values in positive characteristic.

Abstract

We study algebraic independence problem for the Taylor coefficients of the Anderson-Thakur series arisen as deformation series of positive characteristic multiple zeta values (abbreviated as MZV's). These Taylor coefficients are simply specialization of hyperderivatives of the Anderson-Thakur series. We consider the prolongation of t-motives associated with MZV's, and then determine the dimension of the t-motivic Galois groups in question under certain hypothesis. By using Papanikolas' theory, it enables us to obtain the desired algebraic independence result.