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Wagner, Sven
On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields. Annales de la faculté des sciences de Toulouse Mathématiques, Sér. 6, 19 no. S1 (2010), p. 221-242
Analyses MR 2675729 | Zbl 1210.14069
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Résumé

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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