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On the substitution theorem for rings of semialgebraic functions

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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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Official URL: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9295148&fileId=S1474748014000206


URLURL Type
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
Uncontrolled Keywords:semialgebraic set; ring of semialgebraic functions; extension of coefficients; evaluation homomorphisms; substitution theorem; weak continuous extension property
Subjects:Sciences > Mathematics
Sciences > Mathematics > Algebra
ID Code:29130
Deposited On:10 Mar 2015 09:37
Last Modified:10 Mar 2015 09:37

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