Baro González, ElíasFernando, JoséGamboa Mutuberria, José Manuel2023-06-172023-06-172018Baro, 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.https://hdl.handle.net/20.500.14352/13084In 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.engjournal articleopen access514512515.1ÁlgebraTopologíaGeometríaSemialgebraic compactificationReal closed ringMatemáticas (Matemáticas)ÁlgebraGeometríaTopología12 Matemáticas1201 Álgebra1204 Geometría1210 Topología