$\mathbb{F}_q$-linear relations among Thakur's multiple zeta values in positive characteristic

Abstract

Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(\theta^{-1})\!)$ spanned by Thakur's multiple zeta values $\zeta_A(\mathfrak{s})$ of weight $w$. We prove that $\sum_{w=1}^{\infty} \left(\dim_{\mathbb{F}_q} \mathcal{Z}_w^{(\mathbb{F}_q)}\right) x^w = \frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1})^2}$. Moreover, we construct an explicit $\mathbb{F}_q$-basis of $\mathcal{Z}_w^{(\mathbb{F}_q)}$, and prove that any $\mathbb{F}_q$-linear relation among Carlitz multiple polylogarithm values $\operatorname{Li}_A(\mathfrak{s})$ is an $\mathbb{F}_q$-linear combination of quadruple-carry relations. This result can be regarded as an $\mathbb{F}_q$-analogue of the corresponding $\mathbb{F}_q(\theta)$-theorem proved by Chang--Chen--Mishiba and independently by Im--Kim--Le--Ngo Dac--Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im--Kim--Ngo Dac. These relations may be viewed as $\mathbb{F}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta(\mathfrak{s})$.