[Paper Review] Definable retractions over complete fields with separated power series

Krzysztof Jan Nowak|arXiv (Cornell University)|Jan 1, 2019

Advanced Topology and Set Theory14 references4 citations

TL;DR

This paper establishes the existence of definable retractions onto any closed definable subset in complete non-Archimedean fields with separated power series, using tools like embedded resolution of singularities, elimination of valued field quantifiers, and quasi-rational subdomains. The key result is a non-Archimedean definable analogue of the Tietze–Urysohn and Dugundji extension theorems in the Denef–Pas language framework.

ABSTRACT

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.

Motivation & Objective

• To establish definable retractions onto closed definable subsets in complete non-Archimedean fields with separated power series.
• To provide a non-Archimedean definable analogue of the classical Tietze–Urysohn and Dugundji extension theorems.
• To reduce the problem of definable retractions to the case of simple normal crossing divisors via desingularization and closedness theorems.
• To apply model-theoretic tools such as elimination of valued field quantifiers and the analytic closedness theorem to non-Archimedean geometry.

Proposed method

• Reduction of the retraction problem to simple normal crossing divisors using the closedness theorem and desingularization of terms.
• Application of embedded resolution of singularities via blowing up in quasi-compact rigid analytic spaces.
• Utilization of quasi-rational subdomains to control definable sets and their topological structure.
• Employment of elimination of valued field quantifiers in the Denef–Pas language to simplify definable conditions.
• Construction of retractions through inductive gluing over clopen coverings of quasi-rational subdomains.
• Use of the analytic closedness theorem to ensure compatibility of resolution with the canonical topology induced by the valuation.

Experimental results

Research questions

• RQ1 Can definable retractions exist onto any closed definable subset in complete non-Archimedean fields with separated power series?
• RQ2 How can the Tietze–Urysohn and Dugundji extension theorems be generalized in a definable, non-Archimedean setting?
• RQ3 What role do simple normal crossing divisors play in reducing the general retraction problem to a manageable case?
• RQ4 How do model-theoretic tools like elimination of quantifiers and the closedness theorem interact with geometric resolution techniques?
• RQ5 To what extent can definable retractions be constructed in products of unit balls and projective spaces over such fields?

Key findings

• An $ι$-definable retraction exists from any closed $ι$-definable subset $A$ of $K^n$ to $A$, as stated in Corollary 1.2.
• The existence of definable retractions onto closed definable subsets in $(K^\circ)^N$, $(K^{\circ\circ})^N$, $\mathbb{P}^n(K)$, and their products is established.
• The retraction problem reduces to the case of simple normal crossing divisors via embedded resolution of singularities and the closedness theorem.
• Definable retractions exist even when the ambient space is the complement of a closed subvariety $Z$, provided $Z$ and the resolution divisors form a simultaneous simple normal crossing configuration.
• The proof relies on inductive construction of retractions over clopen coverings of quasi-rational subdomains, with gluing via homeomorphisms and definable maps.
• The results extend to general Henselian valued fields with analytic structure, as shown in subsequent work [13], confirming broad applicability.