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

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

2008-10-27

arxiv.org/abs/0810.4800v3

Mathematics

Algebraic Geometry

v2: Removed typos, changed content. v3: Added missing conditions for several results in section 6

Scientific paper

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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