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On the Pierce-Birkhoff conjecture for smooth affine surfaces over real closed fields. (English) Zbl 1210.14069
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)].

##### MSC:
 14P10 Semialgebraic sets and related spaces 13J25 Ordered rings
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##### References:
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