×

On the Pierce-Birkhoff conjecture over ordered fields. (English) Zbl 0715.14047

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 \(n=2\) [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.
Reviewer: M.Coste

MSC:

14P10 Semialgebraic sets and related spaces
14P05 Real algebraic sets
41A10 Approximation by polynomials
06F25 Ordered rings, algebras, modules

Citations:

Zbl 0578.41008
PDFBibTeX XMLCite
Full Text: DOI