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Rings of differentiable semialgebraic functions

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Baro, Elías and F. Fernando, José and Gamboa, José Manuel (2018) Rings of differentiable semialgebraic functions. (Unpublished)

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Abstract

In this work we analyze the main properties of the Zariski and maximal spectra of a ring S^r(M) of differentiable semialgebraic functions of class C^r on a semialgebraic subset M of R^m where R denotes the field of real numbers. Denote S^0(M) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in Cl(M), which is the real closure of S^r(M) . Despite S^r(M) it is not real closed for r>0, the Zariski and maximal spectra are homeomorphic to the corresponding ones of the real closed ring S^0(M). Moreover, we show that the quotients of S^r(M) by its prime ideals have real closed fields of fractions, so the ring S^r(M) is close to be real closed. The equality between the spectra of S^r(M) and S^0(M) guarantee that the properties of these rings that depend on such spectra coincide. For instance the ring S^r(M) is a Gelfand ring and its Krull dimension is equal to dim(M). If M is locally compact, the ring S^r(M) enjoys a Nullstellensatz result and Lojasiewicz inequality. We also show similar results for the ring of differentiable bounded semialgebraic functions.


Item Type:Article
Uncontrolled Keywords:Álgebra, Topología, Geometría
Palabras clave (otros idiomas):Semialgebraic compactification, Real closed ring
Subjects:Sciences > Mathematics
Sciences > Mathematics > Algebra
Sciences > Mathematics > Geometry
Sciences > Mathematics > Topology
ID Code:51183
Deposited On:13 Feb 2019 12:24
Last Modified:13 Feb 2019 12:24

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