Approximate roots of a valuation and the Pierce-Birkhoff conjecture
type: Item
instance of: scholarly article
title: Approximate roots of a valuation and the Pierce-Birkhoff conjecture
zbMATH Open document ID: 1271.13051
author: M. Spivakovsky
author: F. Lucas
author: J. Madden
author: D. Schaub
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2012-08-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.1188
reviewed by: Igor Klep
Mathematics Subject Classification ID: 13J30
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 6062792
zbMATH Keywords: valuation
zbMATH Keywords: regular local ring
zbMATH Keywords: approximate root
zbMATH Keywords: Pierce-Birkhoff conjecture
zbMATH Keywords: connectedness conjecture
MaRDI profile type: Publication
OpenAlex ID: W2963184641
Wikidata QID: Q123267472
arXiv ID: 1003.1188
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DOI: 10.5802/AFST.1336
review text: This paper is concerned with the Pierce-Birkhoff conjecture: any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. The Pierce-Birkhoff conjecture was proved by \textit{L. Mahé} in 1984 for \(n\leq2\) [Rocky Mt. J. Math. 14, 983--985 (1984; Zbl 0578.41008)] but is still open for \(n>2\).NEWLINENEWLINEIn the article under review the authors continue their quest for a real algebro-geometric proof of the conjecture via their Connectedness conjecture from [\textit{F. Lucas, J. J. Madden, D. Schaub} and \textit{M. Spivakovsky}, Manuscr. Math. 128, No. 4, 505--547 (2009; Zbl 1169.14039)]. As the main tool they use approximate roots of a regular local ring \((A,{\mathbf m})\) equipped with a valuation centered in \({\mathbf m}\). They establish a definable connectedness conjecture for \(\dim A=2\), leading to a new proof of the Mahé theorem cited above.
description: scientific article; zbMATH DE number 6062792