A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.
Nash structure of curves over a valued field and tame henselian rationality
Sorbonne Université, Université Paris-Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG（索邦大学，巴黎第七大学，法国国家科学研究中心，儒勒·让苏-巴黎左岸数学研究所）；Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS（索邦大学，儒勒·让苏-巴黎左岸数学研究所，法国国家科学研究中心）
Elimination results for tame fields with finite residue fields
Université Paris Cité, Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学、索邦大学、法国国家科学研究中心、IMJ-PRG）；Heinrich Heine University Düsseldorf, Faculty of Mathematics and Natural Sciences（杜塞尔多夫海因里希·海涅大学、数学与自然科学学院）
本文基于 Kuhlmann 和 Lisinski 的工作，研究有限域上的 Hahn 系列场的理论，证明每个公式等价于存在 y 使得多项式 f(x₁,…,xₙ,y)=0。
