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Preprint Number 1931
1931. Krzysztof Jan Nowak
Tame topology and desingularization in Hensel minimal structures
Submission date: 2 March 2021. Revised version, 31 August 2021, with new title and abstract.
This paper deals with Hensel minimal, non-trivially valued fields K of equicharacteristic zero, whose axiomatic theory was introduced in a recent article by Cluckers-Halupczok-Rideau. We additionally require that the standard algebraic language be induced (up to interdefinability) for the imaginary sort RV . This condition is satisfied by the majority of classical tame structures on Henselian fields, including Henselian fields with analytic structure. The main purpose is to carry over many results of our previous papers to the above general axiomatic settings including, among others, the theorem on existence of the limit, curve selection, the closedness theorem, several non-Archimedean versions of the Lojasiewicz inequalities as well as the theorems on extending continuous definable functions and on existence of definable retractions. We establish an embedding theorem for regular definable spaces and the definable ultranormality of definable Hausdorff LC-spaces. Also given are examples that curve selection and the closedness theorem, key results for numerous applications, may be no longer true after expanding the language for the leading term structure RV . In the case of Henselian fields with analytic structure, a more precise version of the theorem on existence of the limit (a version of Puiseux's theorem) is provided. Further, we establish definable versions of resolution of singularities (hypersurface case) and transformation to normal crossings by blowing up, on arbitrary strong analytic manifolds in Hensel minimal expansions of analytic structures. Also introduced are meromorphous functions, i.e. continuous quotients of strong analytic functions on strong analytic manifolds. Finally, we prove a finitary meromorphous version of the Nullstellensatz.
Mathematics Subject Classification: 12J25, 14G22, 14G27, 03C10, 32P05, 32B05
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