Universal $\mathbb{F}_q-$Family of $v-$adic Multiple Zeta Values over Function Fields

TL;DR

This paper establishes a universal family of linear relations among interpolated $v$-adic multiple zeta values over function fields, potentially generating all such relations and extending to multiple harmonic sums.

Contribution

It introduces a universal family of linear relations for interpolated $v$-adic MZVs over function fields, a conjecture that these generate all relations.

Findings

Proved a universal family of linear relations for interpolated $v$-adic MZVs.

Conjectured these relations generate all linear relations over $ ext{F}_q$.

Extended the relations to all multiple harmonic type sums.

Abstract

This paper aims to study the $\mathbb{F}_q-$linear relations between interpolated $v-$adic multiple zeta values over function fields. We proved a universal family of linear relations of interpolated $v-$adic MZVs, which is conjectured to generate all linear relations over $\mathbb{F}_q$. At the end of the paper, we also show that these relations can be generalized to all multiple harmonic type sums.