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Basics of O-minimality and Hardy Fields

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Book cover Lecture Notes on O-Minimal Structures and Real Analytic Geometry

Part of the book series: Fields Institute Communications ((FIC,volume 62))

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.

Mathematics Subject Classification (2010): Primary 03C64, Secondary 26A12

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Notes

  1. 1.

    Often called “0-definable” in the model-theoretic literature.

  2. 2.

    In many older papers in model theory, the default is that “definable” means “∅-definable”.

  3. 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. 4.

    A proper expansion of \(\overline{\mathbb{R}}\) is one that defines a non-semialgebraic set.

  5. 5.

    That is, its theory is axiomatizable by universal sentences.

  6. 6.

    Exercise. Prove it.

  7. 7.

    The argument is essentially from van den Dries et al. [38].

  8. 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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  1. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH, 43210, USA

    Chris Miller

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  1. , Department of Mathematics, The Ohio State University, West 18th Avenue 231, Columbus, 43210, Ohio, USA

    Chris Miller

  2. , Institut de mathématiques de Bourgogne, Université de Bourgogne, avenue Alain Savary 9, Dijon, 21078, France

    Jean-Philippe Rolin

  3. , Department of Mathematics and Statistics, McMaster University, Main Street West 1280, Hamilton, L8S 4K1, Ontario, Canada

    Patrick Speissegger

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Miller, C. (2012). Basics of O-minimality and Hardy Fields. In: Miller, C., Rolin, JP., Speissegger, P. (eds) Lecture Notes on O-Minimal Structures and Real Analytic Geometry. Fields Institute Communications, vol 62. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4042-0_2

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