A closedness theorem and applications in geometry of rational points over Henselian valued fields

Krzysztof Jan Nowak

Journal of Singularities
volume 21 (2020), 212-233

Received: 2 March 2018. Received in revised form: 12 December 2018.

DOI: 10.5427/jsing.2020.21m

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Abstract:

We develop geometry of algebraic subvarieties of K^n over arbitrary Henselian valued fields K of equicharacteristic zero. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach is again the closedness theorem to the effect that the projections K^n x P^m(K) -> K^n are definably closed maps. It enables, in particular, application of resolution of singularities in much the same way as over locally compact ground fields. As before, the proof of that theorem uses, among others, the local behavior of definable functions of one variable and fiber shrinking, being a relaxed version of curve selection. But now, to achieve the former result, we first examine functions given by algebraic power series. All our previous results will be established here in the general settings: several versions of curve selection (via resolution of singularities) and of the Łojasiewicz inequality (via two instances of quantifier elimination indicated below), extending continuous hereditarily rational functions as well as the theory of regulous functions, sets and sheaves, including Nullstellensatz and Cartan's theorems A and B. Two basic tools are quantifier elimination for Henselian valued fields due to Pas and relative quantifier elimination for ordered abelian groups (in a many-sorted language with imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications of the closedness theorem are piecewise continuity of definable functions, Hölder continuity of functions definable on closed bounded subsets of K^n, the existence of definable retractions onto closed definable subsets of K^n and a definable, non-Archimedean version of the Tietze--Urysohn extension theorem. In a recent paper, we established a version of the closedness theorem over Henselian valued fields with analytic structure along with several applications.

2000 Mathematical Subject Classification:

Primary 12J25, 14B05, 14P10; Secondary 13J15, 14G27, 03C10

Key words and phrases:

valued fields, algebraic power series, closedness theorem, blowing up, descent property, quantifier elimination for Henselian valued fields, quantifier elimination for ordered abelian groups, fiber shrinking, curve selection, Łojasiewicz inequalities, hereditarily rational functions, regulous Nullstellensatz, regulous Cartan's theorems

Author(s) information:

Krzysztof Jan Nowak
Institute of Mathematics
Faculty of Mathematics and
Computer Science
Jagiellonian University
ul. Profesora Łojasiewicza 6
30-348 Kraków, Poland
email: nowak@im.uj.edu.pl