Intermediate algebras of semialgebraic functions

Abstract

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

Accepted for its pubblication