ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS

Abstract

Let$R\subset F$be an extension of real closed fields, and let${\mathcal{S}}(M,R)$be the ring of (continuous) semialgebraic functions on a semialgebraic set$M\subset R^{n}$. We prove that every$R$-homomorphism${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$is essentially the evaluation homomorphism at a certain point$p\in F^{n}$adjacent to the extended semialgebraic set$M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when$M$is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.