Wagner, Sven On the Pierce-Birkhoff conjecture for smooth affine surfaces over real closed fields. (English) Zbl 1210.14069 Ann. Fac. Sci. Toulouse, Math. (6) 19, Spec. Issue, 221-242 (2010). The Pierce-Birkhoff conjecture asserts that if \(h:{\mathbb R}^n \to {\mathbb R}\) is a piecewise polynomial function with finitely many pieces, then it can be written as \[ h = \sup_{1 \leq i \leq r} ( \inf_{1 \leq j \leq s} h_{ij}) \] for some polynomials \(h_{ij}\), that is, \(h\) is the supremum of the infimum of finitely many polynomial functions \(h_{ij}\). For \(n \leq 2\) the conjecture was proved by L. Mahé [Rocky Mountain J. Math. 14, 983–985 (1984; Zbl 0578.41008)]. In this paper the author proves the Pierce-Birkhoff conjecture for non-singular two-dimensional affine real algebraic varieties over real closed fields. In fact, it is proved the so-called Connectedness Conjecture for the coordinate ring of such varieties. The Connectedness Conjecture is concerned with the question whether certain subsets of the real spectrum of the coordinate ring are topologically connected and implies the Pierce-Birkhoff conjecture, as was proved by F. Lucas, J. J. Madden, D. Schaub and M. Spivakovsky [Manuscr. Math. 128, 505–547 (2009; Zbl 1169.14039)]. Reviewer: Antonio Diaz-Cano (Madrid) Cited in 3 Documents MSC: 14P10 Semialgebraic sets and related spaces 13J25 Ordered rings Keywords:Pierce-Birkhoff Conjecture; Connectedness Conjecture; piecewise polynomial functions PDF BibTeX XML Cite \textit{S. Wagner}, Ann. Fac. Sci. Toulouse, Math. (6) 19, 221--242 (2010; Zbl 1210.14069) Full Text: DOI Numdam EuDML arXiv References: [1] 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. · Zbl 0849.13014 [2] J. Bochnak, M. Coste, and M.-F. Roy. Real Algebraic Geometry. Springer-Verlag, 1998. · Zbl 0912.14023 [3] F. Lucas, J. J. Madden, D. Schaub, and M. Spivakovsky. On connectedness of sets in the real spectra of polynomial rings. Preprint, 2007. · Zbl 1169.14039 [4] J. J. Madden. Pierce-Birkhoff rings. Archiv der Mathematik, 53:565 - 570, 1989. · Zbl 0691.14012 [5] L. Mahé. On the Pierce-Birkhoff conjecture. Rocky Mountain Journal of Mathematics, 14:983 - 985, 1984. · Zbl 0578.41008 [6] O. Zariski and P. Samuel. Commutative Algebra, volume II. Van Nostrand, 1960. · Zbl 0121.27801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.