Delzell, Charles N. On the Pierce-Birkhoff conjecture over ordered fields. (English) Zbl 0715.14047 Rocky Mt. J. Math. 19, No. 3, 651-668 (1989). 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 Cited in 3 ReviewsCited in 12 Documents MSC: 14P10 Semialgebraic sets and related spaces 14P05 Real algebraic sets 41A10 Approximation by polynomials 06F25 Ordered rings, algebras, modules Keywords:Pierce Birkhoff conjecture; piecewise polynomial function; piecewise rational functions Citations:Zbl 0578.41008 PDFBibTeX XMLCite \textit{C. N. Delzell}, Rocky Mt. J. Math. 19, No. 3, 651--668 (1989; Zbl 0715.14047) Full Text: DOI