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Preprint Number 1549

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1549. Krzysztof Jan Nowak
Definable retractions onto closed definable subsets over complete fields with separated power series

Submission date: 1 January 2019


Let K be a complete non-Archimedean field K with separated power series, treated in the analytic Denef--Pas language. We prove the existence of definable retractions onto an arbitrary closed definable subset of K^{n}, whereby definable non-Archimedean versions of the extension theorems by Tietze--Urysohn and Dugundji follow directly. We reduce the problem to the case of a simple normal crossing divisor, relying on our closedness theorem and desingularization of terms. The latter result is established by means of the following tools: elimination of valued field quantifiers (due to Cluckers--Lipshitz--Robinson), embedded resolution of singularities by blowing up (due to Bierstone--Milman or Temkin), the technique of quasi-rational subdomains (due to Lipshitz--Robinson) and our closedness theorem.

Mathematics Subject Classification: 32P05, 32B20, 14G22 (Primary), 14P15, 32S45, 03C10

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Full text arXiv 1901.00162: pdf, ps.

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