$b$-minimality

R. Cluckers; F. Loeser

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$b$-minimality

R. Cluckers·F. Loeser

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Abstract

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. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of P. J. Cohen, J. Denef, and the link between cell decomposition and integration was first made by Denef.

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BibTeX RIS

R. Cluckers, F. Loeser. 2006-10-05. $b$-minimality. https://arxiv.org/abs/math/0610183

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