Henselian valued fields的存在理论在形式光滑性假设下
Existential theories of henselian valued fields under a formal smoothness assumption
AI总结：
本文在形式光滑性假设下研究Henselian赋值域的存在理论，并探讨其在函数域渐进行为理论中的应用。
摘要：
我们研究了正特征Henselian赋值域的存在理论，参数来自一个平凡赋值子域。与之前的工作相比，我们放宽了完美性和分离性假设，转而使用更弱的代数假设，即在参数域上形式光滑性，我们在本设定中对此进行了详细讨论。假设一个解析奇点的弱后果，该假设已在之前的工作中使用过，我们以残差域的存在理论来刻画此类赋值域的存在理论，两者都在同一参数域上。这一结果自然应用于曲线函数域的渐进行为理论。我们详细讨论了函数域在相对较一般伪代数闭合域上的情况，其中我们获得了所有完成或所有但有限多个完成中成立的全称/存在句子集的可判定性。
英文摘要：
We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the weaker algebraic hypothesis of formal smoothness over the parameter field, which we discuss in detail in our setting. Assuming a weak consequence of resolution of singularities, which was already used in previous work, we obtain an axiomatisation of the existential theory of such a valued field in terms of the existential theory of the residue field, both over the same parameter field. This result has natural applications to asymptotic theories of completions of function fields of curves. We work these out in detail for the case of function fields over fairly general pseudo-algebraically closed fields, where we obtain decidability of the sets of universal/existential sentences holding in all completions or all but finitely many completions, respectively.
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