# Fields with analytic structure

@article{Cluckers2009FieldsWA, title={Fields with analytic structure}, author={R. Cluckers and L. Lipshitz}, journal={Journal of the European Mathematical Society}, year={2009}, volume={13}, pages={1147-1223} }

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principle, previously studied, analytic structures on Henselian valued fields, as… Expand

#### 65 Citations

Real closed fields with non-standard and standard analytic structure

- Mathematics
- 2008

We consider the ordered field which is the completion of the Puiseux series field over equipped with a ring of analytic functions on [�1, 1]n which contains the standard subanalytic functions as well… Expand

Some results of geometry over Henselian fields with analytic structure

- Mathematics
- 2018

The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as… Expand

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

- Mathematics, Computer Science
- Symmetry
- 2019

The main purpose is to give a definable version of the canonical desingularization algorithm due to Bierstone--Milman so that both these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. Expand

A closedness theorem over Henselian fields with analytic structure and its applications

- Mathematics
- 2020

In this brief note, we present our closedness theorem in geometry over Henselian valued fields with analytic structure. It enables, among others, application of resolution of singularities and of… Expand

Strictly convergent analytic structures

- Mathematics
- 2013

We give conclusive answers to some questions about definability in analytic languages that arose shortly after the work by Denef and van den Dries, [DD], on $p$-adic subanalytic sets, and we continue… Expand

Effective model-completeness for p-adic analytic structures

- Mathematics
- 2014

In this paper, we combine classical techniques of model theory of p-adic subanalytic sets with results of tropical analytic geometry to obtain a result of effective model-completeness. We consider… Expand

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

- Mathematics
- 2020

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one… Expand

SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS

- Mathematics
- Journal of the Institute of Mathematics of Jussieu
- 2015

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$ -Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$… Expand

A G ] 2 M ar 2 02 1 SOME RESULTS OF GEOMETRY IN HENSEL MINIMAL STRUCTURES

- 2021

We deal with Hensel minimal, non-trivially valued fields K of equicharacteristic zero, whose axiomatic theory was introduced in a recent paper by Cluckers–Halupczok–Rideau. We additionally require… Expand

Definable functions in tame expansions of algebraically closed valued fields

- Mathematics
- 2018

In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: type (I), which hold in a… Expand

#### References

SHOWING 1-10 OF 65 REFERENCES

Real closed fields with non-standard and standard analytic structure

- Mathematics
- 2008

We consider the ordered field which is the completion of the Puiseux series field over equipped with a ring of analytic functions on [�1, 1]n which contains the standard subanalytic functions as well… Expand

The Elementary Theory of Restricted Analytic Fields with Exponentiation

- Mathematics
- 1994

numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the… Expand

Analytic cell decomposition and analytic motivic integration

- Mathematics
- 2005

Abstract The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef–Pas language, and its application to analytic motivic… Expand

MAXIMALLY COMPLETE FIELDS

- 1993

Kaplansky proved in 1942 that among all fields with a valuation having a given divisible value group G, a given algebraically closed residue field R, and a given restriction to the minimal subfield… Expand

Analytic difference rings

- Mathematics
- 2006

Generalizing and synthesizing earlier work on the model theory of valued difference
fields and on the model theory of valued fields with analytic structure, we prove Ax�Kochen�
Er�ov style relative… Expand

Rigid subanalytic sets

- Mathematics
- 1994

Let K be an algebraically closed field endowed with a complete non-archimedean norm. Let f : Y → X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y ) ⊂… Expand

Rings of separated power series and quasi-affinoid geometry

- Mathematics
- 2000

— The papers in this volume present a theory of rigid analytic geometry over an ultrametric field K that generalizes the classical, affinoid, theory to the setting of relative rigid analytic geometry… Expand

Analytic Functions of Several Complex Variables

- Mathematics
- 2011

The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic… Expand

Uniform properties of rigid subanalytic sets

- Mathematics
- 2005

In the context of rigid analytic spaces over a non-Archimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier… Expand

Expansions of algebraically closed fields in o-minimal structures

- Mathematics
- 2001

Abstract. We develop a notion of differentiability over an algebraically closed field K of characteristic zero with respect to a maximal real closed subfield R. We work in the context of an o-minimal… Expand