We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of — and therefore an algorithm to decide — the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses than the algorithm of Denef and Schoutens, which we also discuss in detail. In fact, the consequence of resolution of singularities our results are conditional on is the weakest under which they hold true.

中文翻译：

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𝔽q((t)) 的存在理论公理化

我们使用杰出的均匀化器研究等特征亨塞尔值场的存在理论。特别是，假设奇点解析的结果很弱，我们获得了相对于余数场的存在理论的存在理论的公理化，从而获得了决定的算法。与我们也详细讨论的 Denef 和 Schoutens 算法相比，这更通用，并且在分辨率更弱的假设下工作。事实上，我们的结果所依赖的奇点解决的结果是它们成立的最弱的结果。