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

Exactly the same proof which shows that the Connectedness Property implies the Pierce- Birkhoff Conjecture applies verbatim to show that the Definable Connectedness Property implies

(2) Automorphisms that have positive entropy, also called hyperbolic type au- tomorphisms, can only occur on tori, K3 surfaces, Enriques surfaces, and (nonminimal) rational surfaces,

Finally, we prove the Definable connectedness conjecture and hence the Pierce-Birkhoff conjecture for an arbitrary regular 2-dimensional local ring A; we also prove Conjecture

Une partie de la Conf´ erence 2007 sur les Anneaux Ordonn´ es (les expos´ es de Delzell, Schaub et Spivakovsky et l’article de Wagner publi´ e dans le pr´ esent volume)

— In Remark 5.4 below, we shall use (2.3) above to see that when a piecewise generalized polynomial function h is continuous, each A i in Definition 2.1 can automatically be taken to

The following corollary follows immediately from the fact that an arbitrary algebraic extension field of Q is the direct limit of certain algebraic number fields and from the fact

The exposition of the proof is organized as follows: Section 1 introduces complex multiplication and the polynomials PD, Section 2 describes the factorization of.. PD

Here, the construction of L A/K ∞ , the proof that N K ∞ is torsion and the proof of (3) are simultaneous and rely essentially on the main result of [TV] (a sheaed version of [KT]