On Bröcker's \(t\)-invariant and separating families for constructible sets
type: Item
instance of: scholarly article
title: On Bröcker's \(t\)-invariant and separating families for constructible sets
zbMATH Open document ID: 0829.14026
DOI: 10.1007/BF01832992
author: Murray Marshall
published in: Aequationes Mathematicae
publication date: 1994-12-07T00:00:00Z
full work available at URL: https://eudml.org/doc/137614
review text: The author continues his work in the theory of minimal generation of constructible sets in the real spectrum of a commutative ring. The main result gives upper bounds on the complexity of the constructible sets. Let \(X\) be a saturated and simply layered subset (see the definition in the paper) of the real spectrum of a commutative ring \(A\) of dimension \(d\), and let \(S \subseteq X\) be a constructible set described by a fixed separating family \(f_1, \ldots, f_n\). Then there exist invariants \(s_1, \ldots, s_d\) depending only on \(X\) and on elements \(f_1, \ldots, f_n\) such that (1) \(S\) has a separating family with \(\leq \sum^d_{i = 1} p(s_i)\) elements, where \(p(s) = 4^s - 2^s + 1\). (2) \(S\) is expressible as a union of \(\leq \sum^d_{i = 1} t(s_i)\) basis sets, where \(t(s) = s\) if \(t \leq 2\) and \(t(s)={4^{s - 1} - 2^{s-1} + 1\choose 2^{2s-3}-2^{s-2}+1}\), otherwise. Another result states that if \(d(X) = 0\) then \((X,A^*/T^*)\) is a space of orderings.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 68Q25
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E10
zbMATH DE Number: 699784
zbMATH Keywords: minimal generation of constructible sets
zbMATH Keywords: real spectrum
zbMATH Keywords: separating family
reviewed by: Mieczysław Kula
MaRDI profile type: Publication
cites work: Minimal generation of basic open semianalytic sets
cites work: On the real spectrum of a ring and its application to semialgebraic geometry
cites work: Positivbereiche in kommutativen Ringen
cites work: Minimale Erzeugung von Positivbereichen
cites work: On the separation of basic semialgebraic sets by polynomials
cites work: On basic semialgebraic sets
cites work: Q4693114
cites work: An introduction to real algebra
cites work: Q4208891
cites work: Classification of Finite Spaces of Orderings
cites work: Spaces of Orderings IV
cites work: Q4295696
cites work: Q4892938
cites work: Separating families for semi-algebraic sets
cites work: Stability index of real varieties
description: scientific article; zbMATH DE number 699784