UPC condition in polynomially bounded o-minimal structures
type: Item
instance of: scholarly article
title: UPC condition in polynomially bounded o-minimal structures
zbMATH Open document ID: 1073.32002
author: Rafał Pierzchała
published in: Journal of Approximation Theory
publication date: 2005-02-22T00:00:00Z
review text: One of the purposes of this paper is to generalize the main result of \textit{W. Pawłucki} and \textit{W. Plésniak} in the paper [Math. Ann. 275, 467--480 (1986; Zbl 0579.32020)]. That is, the author proves that UPC condition holds in o-minimal structures generated by some quasi-analytic classes of \(C^\infty\) functions. We recall that UPC is the abbreviation for the terms `uniformly polynomially cuspidal' (the notion of UPC set was introduced by Pawłucki and Plésniak in the reference mentioned before). Moreover, there is given a characterization of bounded and plane UPC sets definable in polynomially bounded o-minimal structures. That is, if \(A\subseteq\mathbb R^2\) is bounded and definable in polynomially bounded o-minimal structures, then \(A\) is UPC if and only if \(A\) is fat (that is \(\overline A=\overline {\text{Int}(A)}\)) and the following condition is satisfied: for each \(a\in\overline A\), \(r>0\) and any connected component \(S\) of the set \(\text{Int}(A)\cap B(a,r)\) such that \(a\in\overline S\) there is a polynomial arc \(\gamma:(0,1)\to S\) such that \(\lim_{t\to 0}\gamma(t)=a\), where \(B(a,r)=\{x\in\mathbb R^2: \| x-a\|<r\}\).
Mathematics Subject Classification ID: 32B20
zbMATH DE Number: 2136631
zbMATH Keywords: o-minimal structures
zbMATH Keywords: Markov inequality
reviewed by: Carles Bivià-Ausina
MaRDI profile type: Publication
full work available at URL: j.jat.2004.10.005
OpenAlex ID: W2063669380
cites work: Semianalytic and subanalytic sets
cites work: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant
cites work: Inégalité de Markov dans les ensembles effiles
cites work: Q4392286
cites work: Geometric categories and o-minimal structures
cites work: The real field with convergent generalized power series
cites work: Exponentiation is Hard to Avoid
cites work: Infinite Differentiability in Polynomially Bounded O-Minimal Structures
cites work: Markov's inequality and \(C^{\infty}\) functions on sets with polynomial cusps
cites work: Extension of $C^{∞}$ functions from sets with polynomial cusps
cites work: Q4831757
cites work: Quasianalytic Denjoy-Carleman classes and o-minimality
DOI: 10.1016/J.JAT.2004.10.005
Recommended article: Markov’s inequality in the o-minimal structure of convergent generalized power series
Recommended article: Markov's inequality and \(C^{\infty}\) functions on sets with polynomial cusps
Recommended article: Valeurs critiques asymptotiques d'une fonction définissable dans une structure o-minimale
Recommended article: Extension of \(C^\infty\) functions in polynomially bounded o-minimal structure
Recommended article: Markov inequality on sets with polynomial parametrization
Recommended article: Polynomial inequalities on general subsets of $R^N$
Recommended article: Integer-valued definable functions in ℝan,exp
Recommended article: Q4831757
Recommended article: An o-minimal structure which does not admit \(\mathcal C^{\infty}\) cellular decomposition
Recommended article: Lipschitz stratifications in o-minimal structures
description: scientific article; zbMATH DE number 2136631