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On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields

Sven Wagner — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if $W$ is such a variety, then every piecewise polynomial function on $W$ can be written as suprema of infima of polynomial functions on $W$ . More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.

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