# B-Minimality

@article{Cluckers2006BMinimality, title={B-Minimality}, author={Raf Cluckers and François Loeser}, journal={J. Math. Log.}, year={2006}, volume={7} }

. — We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b -minimality and is based on deﬁnable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b -minimal structures. We show that b -minimality applies to the theory of Henselian valued ﬁelds of characteristic zero, generalizing work by Denef - Pas [ 25 ] [ 26 ]. Structures which are o -minimal, v -minimal, or p -minimal and…

## 36 Citations

### Tame topology and desingularization in Hensel minimal structures

- Mathematics
- 2021

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

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

- Mathematics
- 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…

### Non-archimedean stratifications in T-convex fields.

- Mathematics
- 2017

We prove that whenever T is a power-bounded o-minimal theory, t-stratifications exist for definable maps and sets in T-convex fields. To this effect, a thorough analysis of definability in T-convex…

### 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…

### A version of p-adic minimality

- Mathematics, Computer ScienceThe Journal of Symbolic Logic
- 2012

A very weak language on p-adic fields K is introduced, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language, and it is proved that the only definable functions in the language are trivial functions.

### Motivic local density

- Mathematics
- 2015

We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers–Comte–Loeser about p-adic local density. We prove some results about geometry of definable…

### An introduction to b-minimality

- Mathematics, Philosophy
- 2006

We give a survey with some explanations but no proofs of the new notion of b-minimality by the author and F. Loeser [b-minimality, J. Math. Log., 7 no. 2 (2007) 195--227, math.LO/0610183]. We compare…

### A note on tropicalization in the context of Berkovich spaces

- Mathematics
- 2015

Let K be a non-Archimedean non-trivially valued field, and Γ be its value group. Let X be a compact strictly K-analytic space and f: X → $$A_K^{n,an}$$ be a morphism of K-analytic spaces. We prove…

### Motivic local density

- MathematicsMathematische Zeitschrift
- 2016

We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers–Comte–Loeser about p-adic local density. We prove some results about geometry of definable…

### NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT

- MathematicsForum of Mathematics, Pi
- 2015

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor…

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We give a survey with some explanations but no proofs of the new notion of b-minimality by the author and F. Loeser [b-minimality, J. Math. Log., 7 no. 2 (2007) 195--227, math.LO/0610183]. We compare…

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