A polynomial basis for the multizeta algebra in positive characteristic

Source abstract

Let $K=\mathbb{F}_q(θ)$, and let $\mathcal{Z}$ be the $K$-algebra generated by Thakur's multiple zeta values $ζ_A(\mathfrak{s})$. We prove that $\mathcal{Z}$ is a polynomial algebra and construct an explicit polynomial basis of $\mathcal{Z}$ over $K$. We also determine the transcendence degree of $K[ζ_A(\mathfrak{s}): \operatorname{wt}(\mathfrak{s}) \leqslant w]$ over $K$ for every integer $w \geqslant 1$, generalizing a result of Ngo Dac--Nguyen Chu--Pham. The proof uses Chang's grading theorem, the linear basis theorem proved independently by Chang--Chen--Mishiba and Im--Kim--Le--Ngo Dac--Pham, and the carry relations of Im--Kim--Ngo Dac. The key step is to show that suitable derivations obtained from deconcatenation and linear functionals descend through the carry relations to $\mathcal{Z}$.