Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic

Abstract: We prove Todd's relation conjecture: all $\mathbb{F}<em>q(θ)$-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators $\mathcal{B}<sup>{\mathrm{S}}$, $\mathcal{C}<sup>{\mathrm{S}}$, $$\mathcal{B}^{{\mathrm{S}}\circ} \mathcal{C}<sup>{\mathrm{S}}$$; moreover they are also generated by $\mathcal{B}<sup>{\ast\mathrm{S}}$, $\mathcal{C}<sup>{\mathrm{S}}$, $$\mathcal{B}<sup>{\ast\mathrm{S}}\circ\mathcal{C}<sup>{\mathrm{S}}$$. The $\mathcal{B}<sup>{\ast\mathrm{S}}$-part of this conjecture has been proved by Chang, Chen and Mishiba. We prove the whole conjecture for Carlitz multiple polylogarithm values, which implies the $\mathcal{B}<sup>{\mathrm{S}}$-part for multiple zeta values. We also determine all fixed relations and binary relations. Let $\mathfrak{BR}<sup>{\mathrm{S}}_w$ be the $\mathbb{F}_q(θ)$-linear space spanned by binary relations of weight $w$, and let $\operatorname{Fix}<sup>{\mathrm{S}}_w$ be the $\mathbb{F}_q(θ)$-linear space spanned by fixed relations. We derive generating functions $$\sum</em>{w\ge 1}\bigl(\dim_{\mathbb{F}<em>q(θ)}\operatorname{Fix}<sup>{\mathrm{S}}_w\bigr)x<sup>w=\frac{x<sup>{q+1}(1-x)}{(1-2x)(1-2x+x<sup>{q+1})}$$ and $$\sum</em>{w\ge 1}\bigl(\dim_{\mathbb{F}_q(θ)}\mathfrak{BR}<sup>{\mathrm{S}}_w\bigr)x<sup>w=\frac{x<sup>q(1-x)}{(1-2x)(1-2x+x<sup>{q+1})}.$$ Our results are based on the recent work of Im-Kim-Ngo Dac on the $\mathbb{F}_q$-linear relations of Thakur's multiple zeta values, and a system of transfer theorems between multiple zeta values and multiple polylogarithm values.