# On the substitution theorem for rings of semialgebraic functions

### Impacto

Fernando Galván, José Francisco (2014) On the substitution theorem for rings of semialgebraic functions. Journal of the Institute of Mathematics of Jussieu . ISSN 1474-7480

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http://arxiv.org/abs/1309.3743UNSPECIFIED

## Abstract

Let R⊂F be an extension of real closed fields and S(M,R) the ring of (continuous) semialgebraic functions on a semialgebraic set M⊂Rn. We prove that every R-homomorphism φ:S(M,R)→F is essentially the evaluation homomorphism at a certain point p∈Fn \em adjacent \em to the extended semialgebraic set MF. This type of result is commonly known in Real Algebra as Substitution Theorem. In case 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, \em appropriate immersion \em of semialgebraic sets) that have interest on their own. We afford the same problem for the ring of bounded (continuous) semialgebraic functions getting results of a different nature.

Item Type: Article semialgebraic set; ring of semialgebraic functions; extension of coefficients; evaluation homomorphisms; substitution theorem; weak continuous extension property Sciences > MathematicsSciences > Mathematics > Algebra 29130 10 Mar 2015 09:37 10 Mar 2015 09:37