On the Pierce-Birkhoff conjecture over ordered fields
type: Item
instance of: scholarly article
title: On the Pierce-Birkhoff conjecture over ordered fields
zbMATH Open document ID: 0715.14047
DOI: 10.1216/RMJ-1989-19-3-651
author: Charles N. Delzell
published in: Rocky Mountain Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The Pierce Birkhoff conjecture says that a continuous piecewise polynomial function on \({\mathbb{R}}^ n\) (with a finite number of pieces) is a sup of infs of finitely many polynomials. This has been proved only up to \(n=2\) [\textit{L. Mahé}, Rocky Mt. J. Math. 14, 983-985 (1984; Zbl 0578.41008)]. This note contains an improvement on Mahé's result (for \(n=2):\) if the coefficients of the polynomials, which coincide with the function on the pieces, are in a subfield K, then those of the polynomials appearing in the sup of infs may also be taken in K. There are also in the note interesting examples and remarks, notably about piecewise rational functions.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 4177286
zbMATH Keywords: Pierce Birkhoff conjecture
zbMATH Keywords: piecewise polynomial function
zbMATH Keywords: piecewise rational functions
Wikidata QID: Q122860395
reviewed by: Michel Coste
MaRDI profile type: Publication
full work available at URL: rmj-1989-19-3-651
OpenAlex ID: W2006534418
description: scientific article; zbMATH DE number 4177286