A note on a separation problem
type: Item
instance of: scholarly article
title: A note on a separation problem
zbMATH Open document ID: 0547.14017
DOI: 10.1007/BF01193850
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: The subject of this note is the following problem: Let \(A_ 0\), \(Z_ 0\) be semianalytic closed germs in \({\mathbb{R}}^ n_ 0\) with \(A_ 0\cap Z_ 0=\{0\}\). Under what conditions does there exist an analytic function germ h, positive on \(Z_ 0\backslash\{0\}\) and negative on \(A_ 0\backslash\{0\}?\) It is proven that a sufficient condition is dim \(A_ 0=1\), and in that case h can be chosen polynomial. This is applied to solve in low dimensions some open problems on semialgebraic sets. For instance, it is shown: (I) Let U be a 1- or 2-dimensional open semialgebraic set of a real algebraic variety. If U is basic, so is \(\bar U\). - (II) Let V be a real algebraic curve and U an open semialgebraic subset of V. Then \(U=\{h>0\}, \bar U=\{h\geq 0\}\) for some polynomial \(h\in {\mathbb{R}}[V]\). (III) Let V be a real algebraic surface and \(V_ c\) its maximum dimension locus. Then \(V_ c=\{h\geq 0\}\) for a polynomial \(h\in {\mathbb{R}}[V]\).
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 3871541
zbMATH Keywords: separation problem
zbMATH Keywords: semianalytic germ
zbMATH Keywords: analytic function germ
zbMATH Keywords: semialgebraic sets
author: Jesús M. Ruiz
MaRDI profile type: Publication
cites work: Q4101953
cites work: Sur le théorème des fonctions composées différentiables
cites work: Central orderings in fields of real meromorphic function germs
full work available at URL: https://doi.org/10.1007/bf01193850
OpenAlex ID: W2089017372
description: scientific article; zbMATH DE number 3871541