Pierce-Birkhoff rings
type: Item
instance of: scholarly article
title: Pierce-Birkhoff rings
zbMATH Open document ID: 0691.14012
DOI: 10.1007/BF01199816
author: J. Madden
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: The author shows an equivalent formulation of the Pierce-Birkhoff conjecture in terms of the set of separating functions of points of the real spectrum. Precisely, let A denote a commutative ring with unit and SA(A) its ring of abstract continuous semialgebraic functions. Set PW(A)\(\subset SA(A)\) for the subring of functions which are piecewise defined by elements of A. Then it is shown that the following are equivalent: (a) A verifies the Pierce-Birkhoff conjecture, that is, any \(t\in PW(A)\) is of the form \(t=\sup_{i}\{\inf_{j}(b_{ij})\}\) for some \(b_{ij}\in A.\) (b) For any \(t\in PW(A)\) and \(\gamma,\delta \in Spec_ R(A)\), there is \(h\in A\) such that h(\(\gamma)\)\(\geq t(\gamma)\) and h(\(\delta)\)\(\leq t(\delta).\) (c) For any \(t\in PW(A)\) and \(\gamma,\delta \in Spec_ R(A)\), \(t_{\gamma}-t_{\delta}\in <\gamma,\delta >\subseteq A\) is the ideal generated by all functions \(a\in A\) with a(\(\gamma)\)\(\geq 0\) and a(\(\delta)\)\(\leq 0.\) Then the author applies this characterization to show that fields and Dedekind domains satisfy the Pierce-Birkhoff conjecture, improving a result of C. N. Delzell.
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4130538
zbMATH Keywords: polynomial functions
zbMATH Keywords: Pierce-Birkhoff conjecture
zbMATH Keywords: separating functions of points of the real spectrum
zbMATH Keywords: Dedekind domains
reviewed by: Carlos Andradas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4130538