AI 中文总结
本文针对Z_p的分歧扩域A_π，引入π-导子构造算术偏微分算子，证明在温和条件下这类算子与对应解析函数可相互决定，拓展了非分歧情形的相关结论。
AI 中文摘要
Buium引入的p-导子概念在算术几何中具有丰富的应用历史。在$\boldsymbol{\text{Z}_p}$上，Buium-Ralph-Simanca证明，由这些p-导子构造的算术微分算子决定了p进解析函数，反之亦然。值得注意的是，在$\boldsymbol{\text{Z}_p}$上仅存在一个p-导子。本文中，我们对具有一致化子$\boldsymbol{\text{\textbackslash pi}}$的分歧扩域$\boldsymbol{A_\text{\textbackslash pi}}$考虑相同问题。这一工作的效果是从普通微分算子过渡到偏微分算子，因为这类扩域可以拥有多个$\boldsymbol{\text{\textbackslash pi}}$-导子。我们引入一类解析函数，其由这些算子自然决定，方式与非分歧情形类似，但结构丰富得多。我们证明，在温和条件下，该情形中的解析函数与偏微分算子可相互决定。
英文摘要
The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_π$ with uniformizer $π$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $π$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.
