Separating families for semi-algebraic sets
type: Item
instance of: scholarly article
title: Separating families for semi-algebraic sets
zbMATH Open document ID: 0803.14030
DOI: 10.1007/BF03026537
author: Murray Marshall
published in: Manuscripta Mathematica
publication date: 1994-12-15T00:00:00Z
full work available at URL: https://eudml.org/doc/155859
review text: Let \(R\) be the real closure of an ordered field \(F\) and let \(V\) be an algebraic subset of \(\mathbb{R}^ N\) defined over \(F\), i.e., the set of zeros in \(\mathbb{R}^ N\) of some polynomial in \(F[T_ 1, \dots,T_ N]\). For \(a \in \mathbb{R}\) define \(\text{sgn} (a)\) to be 1 (resp. 0, resp. \(-1)\) if \(a>0\) (resp. \(a=0\), resp. \(a<0)\). Obviously, each semialgebraic subset \(S\) of \(V\) defined over \(F\) admits a separating family \(\{f_ 1, \dots, f_ p\}\) of polynomials over \(F\), (that means, for every pair of points \(x \in S\), \(y \in V \backslash S\), there exists an index \(i\) such that \(\text{sgn} (f_ i(x)) \neq \text{sgn} (f_ i(y))\). The paper under review provides estimates of a new invariant \(p_ F(V)\) defined as the least integer such that each semialgebraic set \(S \subset V\) defined over \(F\) admits a separating family consisting of \(\leq p_ F (V)\) polynomials. The main result states the inequality: \[ p_ F(V) \leq \sum^ d_{i = 0} \max \{1,1 + 4^{e + i - 1} - 2^{e + i - 1}\}\tag{1} \] where \(d = \dim V\) and \(e = e_ F \geq 0\) is an invariant of the ordered field \(F\) (e.g., \(e = 1\) if \(F = \mathbb{Q}\), and \(e = 0\) if \(F\) is hereditarily euclidean, for instance if \(F = R)\). On the other hand, for a large class \({\mathcal F}\) of fields -- but distinct of the one consisting of all ordered fields -- the author gets a lower bound for \(p_ F (\mathbb{R}^ d)\). In fact, \({\mathcal F}\) contains \(\mathbb{Q}\) and hereditarily euclidean fields, and if \(\alpha (n) = {n (n + 1) \over 2}\), and \(\beta (n) = 2 \alpha (n)\), it is proved that: \[ \begin{aligned} p_ F (\mathbb{R}^ d) &= \log_ 2 \bigl( \beta^{d - 1} (2) \bigr) + d \quad \text{if} \quad e = 0 \qquad \text{and}\\ p_ F (\mathbb{R}^ d) &= \log_ 2 \biggl( \alpha^{e - 1} \bigl( \beta^ d (2) \bigr) \biggr) + d + e - 1 \quad \text{if} \quad e = 1,\;F \in {\mathcal F}. \end{aligned} \tag{2} \] There are three special cases: \(d=1\), \(e \leq 1\); \(d=2\), \(e=0\), whe…
reviewed by: Jose Manuel Gamboa
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 492807
zbMATH Keywords: semialgebraic set
zbMATH Keywords: separating family
zbMATH Keywords: spaces of orderings
zbMATH Keywords: stability index
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 492807