Zagier-Hoffman's conjectures in positive characteristic

TL;DR

This paper proves the Zagier-Hoffman conjectures in positive characteristic, fully characterizing the linear relations and basis of multiple zeta values in this setting, advancing the understanding of their algebraic structure.

Contribution

It establishes the dimension and explicit basis of multiple zeta values in positive characteristic, confirming the Zagier-Hoffman conjectures.

Findings

All linear relations among alternating multiple zeta values are determined.

The dimension and explicit basis of the span of multiple zeta values are established.

Zagier-Hoffman’s conjectures are fully proved in positive characteristic.

Abstract

Multiples zeta values and alternating multiple zeta values in positive characteristic were introduced by Thakur and Harada as analogues of classical multiple zeta values of Euler and Euler sums. In this paper we determine all linear relations among alternating multiple zeta values and settle the main goals of these theories. As a consequence we completely establish Zagier-Hoffman's conjectures in positive characteristic formulated by Todd and Thakur which predict the dimension and an explicit basis of the span of multiple zeta values of Thakur of fixed weight.