On the algebraic independence of logarithms of Anderson $t$-modules

TL;DR

This paper investigates the algebraic independence of logarithmic coordinates of Anderson t-modules, extending known results on polylogarithms to tensor products of Drinfeld modules and their exterior powers.

Contribution

It establishes new algebraic independence results for logarithms of Anderson t-modules constructed via tensor products and exterior powers, generalizing prior work on polylogarithms.

Findings

Determined algebraic relations among coordinates of logarithms of Anderson t-modules.

Generalized results on algebraic independence to tensor powers of the Carlitz module.

Extended the understanding of algebraic relations in the context of Anderson t-modules.

Abstract

In the present paper, we determine the algebraic relations among the tractable coordinates of logarithms of Anderson $t$-modules constructed by taking the tensor product of Drinfeld modules of rank $r$ defined over the algebraic closure of the rational function field and their $(r-1)$-st exterior powers with the Carlitz tensor powers. Our results, in the case of the tensor powers of the Carlitz module, generalize the work of Chang and Yu on the algebraic independence of polylogarithms.