On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
Sven Wagner[1]
- [1] Universität Konstanz Fachbereich Mathematik und Statistik 78457 Konstanz, Germany
Annales de la faculté des sciences de Toulouse Mathématiques (2010)
- Volume: 19, Issue: S1, page 221-242
- ISSN: 0240-2963
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topWagner, Sven. "On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 221-242. <http://eudml.org/doc/115899>.
@article{Wagner2010,
abstract = {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.},
affiliation = {Universität Konstanz Fachbereich Mathematik und Statistik 78457 Konstanz, Germany},
author = {Wagner, Sven},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Pierce-Birkhoff Conjecture; Connectedness Conjecture; piecewise polynomial functions},
language = {eng},
month = {4},
number = {S1},
pages = {221-242},
publisher = {Université Paul Sabatier, Toulouse},
title = {On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields},
url = {http://eudml.org/doc/115899},
volume = {19},
year = {2010},
}
TY - JOUR
AU - Wagner, Sven
TI - On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 221
EP - 242
AB - 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.
LA - eng
KW - Pierce-Birkhoff Conjecture; Connectedness Conjecture; piecewise polynomial functions
UR - http://eudml.org/doc/115899
ER -
References
top- D. Alvis, B. L. Johnston, and J. J. Madden. Complete ideals defined by sign conditions and the real spectrum of a two-dimensional local ring. Mathematische Nachrichten, 174:21 – 34, 1995. Zbl0849.13014MR1349034
- J. Bochnak, M. Coste, and M.-F. Roy. Real Algebraic Geometry. Springer-Verlag, 1998. Zbl0912.14023MR1659509
- F. Lucas, J. J. Madden, D. Schaub, and M. Spivakovsky. On connectedness of sets in the real spectra of polynomial rings. Preprint, 2007. Zbl1169.14039MR2487439
- J. J. Madden. Pierce-Birkhoff rings. Archiv der Mathematik, 53:565 – 570, 1989. Zbl0691.14012MR1023972
- L. Mahé. On the Pierce-Birkhoff conjecture. Rocky Mountain Journal of Mathematics, 14:983 – 985, 1984. Zbl0578.41008MR773148
- O. Zariski and P. Samuel. Commutative Algebra, volume II. Van Nostrand, 1960. Zbl0121.27801MR120249
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