Intermediate algebras of semialgebraic functions

Abstract

We characterize intermediate ℝ-algebras A between the ring of semialgebraic functions (X) and the ring ∗(X) of bounded semialgebraic functions on a semialgebraic set X as rings of fractions of (X). This allows us to compute the Krull dimension of A, the transcendence degree over ℝ of the residue fields of A and to obtain a Łojasiewicz inequality and a Nullstellensatz for archimedean ℝ-algebras A. In addition we study intermediate ℝ-algebras generated by proper ideals and we prove an extension theorem for functions in such ℝ-algebras.