A theorem of the complement and some new o-minimal structures
type: Item
instance of: scholarly article
title: A theorem of the complement and some new o-minimal structures
zbMATH Open document ID: 0948.03037
author: A. J. Wilkie
published in: Selecta Mathematica. New Series
publication date: 2000-11-19T00:00:00Z
review text: A prestructure is a sequence \((S_n)\) where \(S_n\) is a collection of subsets of \(\mathbb{R}^n\). It is a structure if the \(S_n\) are Boolean algebras, contain the semi-algebraic subsets of \(\mathbb{R}^n\), and for \(m>n\) the projection of \(S_m\) is contained in \(S_n\). A structure is o-minimal if the boundary of every set in \(S_1\) is finite. It is known that \(\mathbb{R}_{\text{an}}\), the structure of globally analytical sets, is o-minimal. Another example of an o-minimal structure is \(\mathbb{R}_{\text{exp}}\), which is generated by the sets of the form \(f^{-1}(0)\), where \(f:\mathbb{R}^n\to\mathbb{R}\) is an exponential polynomial. The author investigates \(\mathbb{R}_{\text{Pfaff}}\), the prestructure of the sets \(f^{-1}(0)\) with \(f:\mathbb{R}^n\to \mathbb{R}\) Pfaffian. He shows that the structure generated by \(\mathbb{R}_{\text{Pfaff}}\) is o-minimal. It is not known whether \(\mathbb{R}_{\text{Pfaff}}\) is closed under complementation.
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 1422751
zbMATH Keywords: Pfaffian function
zbMATH Keywords: o-minimal structure
zbMATH Keywords: prestructure
MaRDI profile type: Publication
full work available at URL: s000290050052
OpenAlex ID: W2093614095
DOI: 10.1007/S000290050052
description: scientific article; zbMATH DE number 1422751