[Paper Review] An introduction to o-minimal structures

Mário J. Edmundo|ArXiv.org|Dec 7, 2000

Advanced Topology and Set Theory9 references3 citations

TL;DR

This paper provides a foundational introduction to o-minimal structures, a framework unifying semialgebraic, subanalytic, and sub-Pfaffian geometries. It establishes key geometric properties of definable sets and maps, demonstrating o-minimality through model-theoretic techniques and highlighting results on model completeness and quantifier elimination in expansions of the real field by analytic and exponential functions.

ABSTRACT

The first papers on o-minimal structures appeared in the mid 1980s, since then the subject has grown into a wide ranging generalisation of semialgebraic, subanalytic and subpfaffian geometry. In these notes we try to show that this is in fact the case by presenting several examples of o-minimal structures and by listing some geometric properties of sets and maps definable in o-minimal structures. We omit here any reference to the pure model theory of o-minimal structures and to the theory of groups and rings definable in o-minimal structures.

Motivation & Objective

• To introduce o-minimal structures as a generalization of semialgebraic, subanalytic, and sub-Pfaffian geometries.
• To present foundational model-theoretic concepts necessary for understanding o-minimal structures.
• To illustrate geometric properties of definable sets and maps in o-minimal structures.
• To survey key examples of o-minimal expansions of the real field, including those with analytic and exponential functions.
• To highlight results on model completeness, quantifier elimination, and cell decomposition in o-minimal theories.

Proposed method

• Using basic model theory to define structures, languages, formulas, and definable sets.
• Applying quantifier elimination and model completeness criteria to analyze definable sets.
• Constructing o-minimal expansions of the real field by adding analytic and exponential functions.
• Employing the Weierstrass preparation theorem and Khovanskii's results to prove o-minimality.
• Utilizing Rolle leaves and the Khovanskii-Rolle theorem in geometric treatments of o-minimality.
• Establishing o-minimal Pfaffian closures and analyzing their model-theoretic properties.

Experimental results

Research questions

• RQ1 What are the defining geometric and model-theoretic properties of o-minimal structures?
• RQ2 How do o-minimal expansions of the real field generalize semialgebraic and subanalytic geometry?
• RQ3 Under what conditions does an expansion of the real field by analytic or Pfaffian functions remain o-minimal?
• RQ4 What is the role of quantifier elimination and model completeness in o-minimal structures?
• RQ5 How do exponential and logarithmic functions affect o-minimality and model completeness in real closed fields?

Key findings

• The structure $\mathbb{R}_{an,exp}$, expanded by restricted analytic functions and the exponential function, is model complete and o-minimal.
• The expansion $\mathbb{R}_{an,exp,log}$ has quantifier elimination and explicit axiomatization.
• The gamma function $\Gamma_{|(0,\infty)}$ is not $\mathbb{R}_{an,exp}$-definable, but is definable in $\mathbb{R}_{G,exp}$.
• The Riemann zeta function $\zeta_{|(1,\infty)}$ is definable in $\mathbb{R}_{an^{*},exp}$ via a series representation.
• The o-minimal Pfaffian closure $\mathcal{P}(\mathbb{R})$ of any o-minimal expansion $\mathbb{R}$ exists and preserves analytic cell decomposition and exponential boundedness.
• There is no known example of an exponential o-minimal expansion of $\mathbb{R}$ that is not exponentially bounded.