On the Pierce-Birkhoff conjecture over ordered fields (Q752124)

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.