We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras introduced by the second author. If the number of variables involved is “not large” in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over ${\mathbb{F}}_q$. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of linear relations between Thakur’s multiple zeta values.

中文翻译：

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泰特代数中的平凡多重 zeta 值

我们研究 Tate 代数中的平凡多重 zeta 值。这些是第二作者介绍的泰特代数中多重 zeta 值的具体例子。如果所涉及的变量的数量在论文中以一种精确的方式“不大”，我们可以赋予这组平凡的多重 zeta 值在非交换多项式环上的模结构，其系数为有理分数${\mathbb{F}}_q$ 上的字段。我们根据生成器确定此模块的结构，并展示在许多情况下，这足以检测 Thakur 的多个 zeta 值之间的线性关系。