RT Journal Article
T1 Rings of differentiable semialgebraic functions
A1 Baro González, Elías
A1 Fernando, José
A1 Gamboa Mutuberria, José Manuel
AB 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.
YR 2018
FD 2018
LK https://hdl.handle.net/20.500.14352/13084
UL https://hdl.handle.net/20.500.14352/13084
LA eng
NO Baro, E., José F. Fernando, y J. M. Gamboa. «Rings of Differentiable Semialgebraic Functions». Selecta Mathematica 30, n.o 4 (septiembre de 2024): 71. https://doi.org/10.1007/s00029-024-00965-z.
DS Docta Complutense
RD 20 abr 2025