2019/01/01 by Nowak, Krzysztof Jan
#03C10 (Secondary) #14G22 (Primary) #14P15 #32B20 #32P05 #32S45 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.00162
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 Kn, 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.