Minimale Erzeugung von Positivbereichen
type: Item
instance of: scholarly article
title: Minimale Erzeugung von Positivbereichen
zbMATH Open document ID: 0546.14016
DOI: 10.1007/BF00147875
published in: Geometriae Dedicata
publication date: 1984-01-01T00:00:00Z
review text: Let \(R\) be a real closed field and \(V\) an affine algebraic \(R\)-variety. A positive cone is a semialgebraic subset \(S\subset V(R)\) of the form \(S=S(f_1,\ldots,f_s)=\{x\in V(R)\mid f_i(x)>0\) for \(i=1,\ldots,s\}\) with \(f_i\in R[V]\). Let \(s(S)=\min\{s\mid S=S(f_1,\ldots,f_s)\}, s(V)=\sup\{s(S)\mid S\) a positive cone in \(V\}\) and \(s(n)=\sup\{s(V)\mid \dim V=n\}\). It is shown that \(s(n)<\infty\). In fact \(s(n)=n\) for \(0<n\le 3\). Also an explicit bound of \(s(n)\) is given for general \(n\) and a characterization of the positive cones among the semialgebraic sets.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E10
zbMATH DE Number: 3869505
zbMATH Keywords: real algebraic geometry
zbMATH Keywords: spaces of orderings
zbMATH Keywords: real spectrum
zbMATH Keywords: positive cone
zbMATH Keywords: semialgebraic subset
author: Ludwig Broecker
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00147875
OpenAlex ID: W74058400
description: scientific article; zbMATH DE number 3869505