Totally odd depth-graded multiple zeta values and period polynomials

TL;DR

This paper investigates the structure and rank of matrices associated with totally odd motivic depth-graded multiple zeta values, providing new proofs and a recursive approach to understanding their relations and conjectures.

Contribution

It offers new proofs for upper bounds on the rank of matrices related to multiple zeta values and introduces a recursive method to analyze their general relations.

Findings

Established new proofs for rank bounds of matrices $C_{N,3}$ and $C_{N,4}$

Proposed a recursive approach reducing the problem to an isomorphism conjecture

Connected the study of multiple zeta values to period polynomials

Abstract

Inspired by a paper of Tasaka, we study the relations between totally odd, motivic depth-graded multiple zeta values. Our main objective is to determine the rank of the matrix $C_{N,r}$ defined by Brown. We will give new proofs for (conjecturally optimal) upper bounds on the rank of $C_{N,3}$ and $C_{N,4}$, which were first obtained by Tasaka. Finally, we present a recursive approach to the general problem, which reduces evaluating the rank of $C_{N,r}$ to an isomorphism conjecture.