On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. S1, p. 221-242
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.
@article{AFST_2010_6_19_S1_221_0,
     author = {Wagner, Sven},
     title = {On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 19},
     number = {S1},
     year = {2010},
     pages = {221-242},
     doi = {10.5802/afst.1283},
     mrnumber = {2675729},
     zbl = {1210.14069},
     language = {en},
     url = {http://http://www.numdam.org/item/AFST_2010_6_19_S1_221_0}
}
Wagner, Sven. On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. S1, pp. 221-242. doi : 10.5802/afst.1283. http://www.numdam.org/item/AFST_2010_6_19_S1_221_0/

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