Approximate roots of a valuation and the Pierce-Birkhoff conjecture
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.
scientific article; zbMATH DE number 6062792
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