AI 中文总结
该研究证明正特征下Thakur多重zeta值子空间的维数生成函数，构造其显式基，明确Carlitz多重多对数值的线性关系结构，给出对应$\boldsymbol{\textit{F}}_\boldsymbol{q}$定理的模拟结果。
AI 中文摘要
设$\boldsymbol{\textit{Z}}_\boldsymbol{w}^{(\boldsymbol{\textit{F}}_\boldsymbol{q})}$是$\boldsymbol{\textit{F}}_\boldsymbol{q}((\theta^{-1}))$中由权为$w$的Thakur多重zeta值$\boldsymbol{\textit{ζ}}_\boldsymbol{A}(\boldsymbol{\textit{s}})$张成的$\boldsymbol{\textit{F}}_\boldsymbol{q}$线性子空间。我们证明$\boldsymbol{\textit{∑}}_{\boldsymbol{w=1}}^\boldsymbol{∞} (\boldsymbol{\textit{dim}}_{\boldsymbol{\textit{F}}_\boldsymbol{q}} \boldsymbol{\textit{Z}}_\boldsymbol{w}^{(\boldsymbol{\textit{F}}_\boldsymbol{q})}) \boldsymbol{\textit{x}}^\boldsymbol{w} = \frac{\boldsymbol{\textit{x}}(\boldsymbol{1}-\boldsymbol{x}^\boldsymbol{q})(\boldsymbol{1}-\boldsymbol{2x}+\boldsymbol{x}^\boldsymbol{q})}{(\boldsymbol{1}-\boldsymbol{2x}+\boldsymbol{x}^{\boldsymbol{q+1}})^\boldsymbol{2}}$；还构造了$\boldsymbol{\textit{Z}}_\boldsymbol{w}^{(\boldsymbol{\textit{F}}_\boldsymbol{q})}$的显式$\boldsymbol{\textit{F}}_\boldsymbol{q}$基，证明Carlitz多重多对数值$\boldsymbol{\textit{Li}}_\boldsymbol{A}(\boldsymbol{\textit{s}})$间的任意$\boldsymbol{\textit{F}}_\boldsymbol{q}$线性关系都是四重进位关系的$\boldsymbol{\textit{F}}_\boldsymbol{q}$线性组合。该结果可视为Chang-Chen-Mishiba与Im-Kim-Le-Ngo Dac-Pham分别证明的对应$\boldsymbol{\textit{F}}_\boldsymbol{q}(\boldsymbol{\textit{θ}})$定理的$\boldsymbol{\textit{F}}_\boldsymbol{q}$模拟，四重进位关系的发现受Im-Kim-Ngo Dac近期工作启发，可看作经典多重zeta值$\boldsymbol{\textit{ζ}}(\boldsymbol{\textit{s}})$间双洗牌关系的$\boldsymbol{\textit{F}}_\boldsymbol{q}$模拟。
英文摘要
Let $\mathcal{Z}_w^{(\mathbb{F}_q)}$ be the $\mathbb{F}_q$-linear subspace of $\mathbb{F}_q(\!(θ^{-1})\!)$ spanned by Thakur's multiple zeta values $ζ_A(\mathfrak{s})$ 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 $\mathbb{F}_q(θ)$-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 $ζ(\mathfrak{s})$.
Comments40 pages, 2 figures
