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ON THE SUBSTITUTION THEOREM FOR RINGS OF SEMIALGEBRAIC FUNCTIONS

Part of: Maps and general types of spaces defined by maps Real and complex fields Chain conditions, finiteness conditions

Published online by Cambridge University Press:  01 July 2014

José F. Fernando
José F. Fernando*
Affiliation:
Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain (josefer@mat.ucm.es)
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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.

Keywords

semialgebraic setring of semialgebraic functionsextension of coefficientsevaluation homomorphismssubstitution theoremweak continuous extension property

MSC classification

Primary: 14P10: Semialgebraic sets and related spaces 54C30: Real-valued functions
Secondary: 12D15: Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 13E99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 4 , October 2015 , pp. 857 - 894
Copyright
© Cambridge University Press 2014 

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