# Definable transformation to normal crossings over Henselian fields with separated analytic structure

## TL;DR

This paper develops a definable version of the resolution of singularities for non-Archimedean analytic geometry over Henselian fields with separated analytic structure, enabling new model-theoretic and topological applications.

## Contribution

It introduces a definable desingularization algorithm within a flexible category of strong analytic manifolds, bridging rigid analytic geometry and model theory.

## Key findings

- Established a definable resolution of singularities in the non-Archimedean setting.
- Applied model-theoretic compactness to analytic geometry.
- Enabled topological applications like definable retractions.

## Abstract

We are concerned with rigid analytic geometry in the general setting of Henselian fields $K$ with separated analytic structure, whose theory was developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone--Milman so that both these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to $K$. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions.