AI 中文总结
本文证明了正特征下Todd关系猜想，即Thakur多重zeta值的所有线性关系由基本二元关系经特定算子生成，并确定了固定关系和二元关系的维数生成函数。
AI 中文摘要
我们证明了Todd关系猜想：Thakur多重zeta值的所有$\mathbb{F}_q(\theta)$-线性关系均由基本二元关系通过算子$\mathcal{B}^{\mathrm{S}}$、$\mathcal{C}^{\mathrm{S}}$、$\mathcal{B}^{\mathrm{S}}\circ \mathcal{C}^{\mathrm{S}}$生成；此外，它们也由$\mathcal{B}^{\ast\mathrm{S}}$、$\mathcal{C}^{\mathrm{S}}$、$\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$生成。该猜想的$\mathcal{B}^{\ast\mathrm{S}}$部分已由Chang、Chen和Mishiba证明。我们证明了Carlitz多重多对数值的整个猜想，这蕴含了多重zeta值的$\mathcal{B}^{\mathrm{S}}$部分。我们还确定了所有固定关系和二元关系。设$\mathfrak{BR}^{\mathrm{S}}_w$为权重$w$的二元关系张成的$\mathbb{F}_q(\theta)$-线性空间，设$\operatorname{Fix}^{\mathrm{S}}_w$为固定关系张成的$\mathbb{F}_q(\theta)$-线性空间。我们推导出生成函数$\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\operatorname{Fix}^{\mathrm{S}}_w\bigr)x^w=\frac{x^{q+1}(1-x)}{(1-2x)(1-2x+x^{q+1})}$和$\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\mathfrak{BR}^{\mathrm{S}}_w\bigr)x^w=\frac{x^q(1-x)}{(1-2x)(1-2x+x^{q+1})}.$我们的结果基于Im-Kim-Ngo Dac关于Thakur多重zeta值的$\mathbb{F}_q$-线性关系的最新工作，以及多重zeta值与多重多对数值之间的传递定理系统。
英文摘要
We prove Todd's relation conjecture: all $\mathbb{F}_q(θ)$-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators $\mathcal{B}^{\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\mathrm{S}}\circ \mathcal{C}^{\mathrm{S}}$; moreover they are also generated by $\mathcal{B}^{\ast\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$. The $\mathcal{B}^{\ast\mathrm{S}}$-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 $\mathcal{B}^{\mathrm{S}}$-part for multiple zeta values. We also determine all fixed relations and binary relations. Let $\mathfrak{BR}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(θ)$-linear space spanned by binary relations of weight $w$, and let $\operatorname{Fix}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(θ)$-linear space spanned by fixed relations. We derive generating functions $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(θ)}\operatorname{Fix}^{\mathrm{S}}_w\bigr)x^w=\frac{x^{q+1}(1-x)}{(1-2x)(1-2x+x^{q+1})}$ and $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(θ)}\mathfrak{BR}^{\mathrm{S}}_w\bigr)x^w=\frac{x^q(1-x)}{(1-2x)(1-2x+x^{q+1})}.$ 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.
Comments17 pages
