Computations about formal multiple zeta spaces defined by binary extended double shuffle relations

TL;DR

This paper computes the dimensions of formal multiple zeta spaces defined by binary extended double shuffle relations up to weight 22, verifying conjectures and revealing Pascal triangle patterns in their structure.

Contribution

It provides the first extensive computational verification of the dimension conjecture for formal multiple zeta spaces using Gaussian elimination.

Findings

Dimensions follow Pascal triangle patterns in depth-graded spaces

Verification of the dimension conjecture up to weight 22

Computational method using Gaussian forward elimination

Abstract

The formal multiple zeta space we consider with a computer is an $\mathbb{F}_2$-vector space generated by $2^{k-2}$ formal symbols for a given weight $k$, where the symbols satisfy binary extended double shuffle relations. Up to weight $k=22$, we compute the dimensions of the formal multiple zeta spaces, and verify the dimension conjecture on original extended double shuffle relations of real multiple zeta values. Our computations adopt Gaussian forward elimination and give information for spaces filtered by depth. We can observe that the dimensions of the depth-graded formal multiple zeta spaces have a Pascal triangle pattern expected by the Hoffman mult-indices.