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Erschienen in:
Blowings-Up of Vector Fields Kapitel lesen Erstes Kapitel lesen

2012 | OriginalPaper | Buchkapitel

Basics of O-minimality and Hardy Fields

verfasst von : Chris Miller

Erschienen in: Lecture Notes on O-Minimal Structures and Real Analytic Geometry

Verlag: Springer New York

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Abstract

This paper consists of lecture notes on some fundamental results about the asymptotic analysis of unary functions definable in o-minimal expansions of the field of real numbers.

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Vorheriges Kapitel Blowings-Up of Vector Fields
Nächstes Kapitel Construction of O-minimal Structures from Quasianalytic Classes
Fußnoten
1
Often called “0-definable” in the model-theoretic literature.
 
2
In many older papers in model theory, the default is that “definable” means “∅-definable”.
 
3
In other words, the only definable subsets of \(\mathbb{R}\) are those that must be there by virtue of the usual ordering of the real line. Hence, the structure is “order-minimal”, thus accounting for the abbreviation “o-minimal” and the use of a plain text font for the “o”.
 
4
A proper expansion of \(\overline{\mathbb{R}}\) is one that defines a non-semialgebraic set.
 
5
That is, its theory is axiomatizable by universal sentences.
 
6
Exercise. Prove it.
 
7
The argument is essentially from van den Dries et al. [38].
 
8
That is, if \(g: {\mathbb{R}}^{n} \rightarrow \mathbb{R}\) is definable, then there is a finite \(\mathcal{F}\subseteq {\mathcal{T}}_{n}\) such that the graph g is contained in the union of the graphs of the \(f \in \mathcal{F}\).
 
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J. Denef, L. van den Dries, p-adic and real subanalytic sets. Ann. Math. (2) 128, 79–138 (1988)
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Metadaten
Titel
Basics of O-minimality and Hardy Fields
verfasst von
Chris Miller
Copyright-Jahr
2012
Verlag
Springer New York
Buch
Lecture Notes on O-Minimal Structures and Real Analytic Geometry

Print ISBN: 978-1-4614-4041-3

Electronic ISBN: 978-1-4614-4042-0

Copyright-Jahr: 2012

DOI
https://doi.org/10.1007/978-1-4614-4042-0_2