1 results
ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 14 / Issue 4 / October 2015
- Published online by Cambridge University Press:
- 01 July 2014, pp. 857-894
- Print publication:
- October 2015
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- Article
- Export citation
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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.
