Algebraic independence of the Carlitz period and the positive characteristic multizeta values at n and (n,n)

TL;DR

This paper investigates algebraic relations among the Carlitz period and certain multizeta values in positive characteristic, establishing conditions for their algebraic independence over algebraic closures.

Contribution

It proves algebraic independence of the Carlitz period and multizeta values at specific integers under certain conditions, extending understanding of their algebraic relations.

Findings

Algebraic independence when 2n is odd.

Either algebraic independence or simple relations over k.

Conditions for algebraic independence based on divisibility of n.

Abstract

Let $k$ be the rational function field over the finite field of $q$ elements and $\bar{k}$ its fixed algebraic closure. In this paper, we study algebraic relations over $\bar{k}$ among the fundamental period $\widetilde{\pi}$ of the Carlitz module and the positive characteristic multizeta values $\zeta(n)$ and $\zeta(n,n)$ for an "odd" integer $n$, where we say that $n$ is "odd" if $q-1$ does not divide $n$. We prove that either they are algebraically independent over $\bar{k}$ or satisfy some simple relation over $k$. We also prove that if $2n$ is "odd" then they are algebraically independent over $\bar{k}$.