# Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

## Abstract

**:**

## 1. Introduction

- The functions and submanifolds involved in the resolution process, are definable and strong analytic. Consequently, via a model-theoretic compactness argument, the orders of those functions are definable, i.e., their equimultiple loci are finite in number and definable. This enables further analysis of the entries ${\nu}_{r}\left(a\right)$ of the invariant, which is a kind of higher order, rational multiplicity of certain strong analytic functions. Hence and by the canonical character of the process, the successive centers of blowups, being the maximum strata of the desingularization invariant, are definable and strong analytic.
- The entries ${\nu}_{r}\left(a\right)$ can be defined by computations which involve orders of vanishing in suitable local coordinates (independently of their choice) induced by generic affine coordinates of the ambient affine space. Therefore, such computations can be performed through suitable definable families of coordinates induced by affine coordinates. This is of great importance, especially in the absence of definable Skolem functions. Hence ${\nu}_{r}\left(a\right)$ turn out to be definable, i.e., their equimultiple loci are finite in number and definable.
- Making use of the closedness theorem, it is possible to partition each ambient manifold, achieved by blowing up, into a finite number of definable clopen pieces so that, on each of them, both the exceptional hypersufaces (which reflect the history of the process and enable the further construction of the desingularization invariant) and next the successive blowup, can be described in a definable geometric way. This geometric bypass compensates for inability to globally describe the centers of the successive blowups in a purely analytic way, which is caused by lack of good algebraic properties of the rings of global analytic functions.
- The canonical algorithm depends only on the completions of the local rings of analytic function germs at the points of the ambient manifolds. Therefore, finite partitions of those manifold into definable clopen pieces do not affect its output data, although quasi-affinoid structure may change. This legitimizes partitions indicated above.

## 2. Strong Analyticity: Blowups and (Weak) Transforms

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**2.**

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Definition**

**1.**

**Proposition**

**3.**

**Corollary**

**1.**

**Corollary**

**2.**

## 3. Definable Desingularization Algorithm

**Remark**

**3.**

**Theorem**

**1.**

**Remark**

**4.**

**Remark**

**5.**

**Theorem**

**2.**

## 4. Application to the Problem of Definable Retractions

**Theorem**

**3.**

**Corollary**

**3.**

- the technique of quasi-rational and R-subdomains (due to Lipshitz–Robinson [9]);

**Remark**

**6.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Remark**

**7.**

**Theorem**

**4.**

## 5. Intricacies of Non-Archimedean Analytic Geometry

**Example**

**1.**

## Funding

## Conflicts of Interest

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Nowak, K.J.
Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure. *Symmetry* **2019**, *11*, 934.
https://doi.org/10.3390/sym11070934

**AMA Style**

Nowak KJ.
Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure. *Symmetry*. 2019; 11(7):934.
https://doi.org/10.3390/sym11070934

**Chicago/Turabian Style**

Nowak, Krzysztof Jan.
2019. "Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure" *Symmetry* 11, no. 7: 934.
https://doi.org/10.3390/sym11070934