Submission date: 20 August 2019

Abstract:

In this work we analyze the main properties of the Zariski and maximal spectra of the ring S^r(M) of differentiable semialgebraic functions of class C^r on a semialgebraic set MD ⊆ ℝ^m. 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). This ring is the real closure of S^r(M). If M is locally compact, the ring S^r(M) enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite S^r(M) is not real closed for r ≥ 1, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring S^0(M). In addition, 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 missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of S^r(M) and S^0(M) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring S^r(M) is a Gelfand ring and its Krull dimension is equal to dim(M). We also show similar properties for the ring S^{r*}(M) of differentiable bounded semialgebraic functions. In addition, we confront the ring S^∞(M) of differentiable semialgebraic functions of class C^∞ with the ring N(M) of Nash functions on M.

Mathematics Subject Classification: 14P10, 46E25

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Full text arXiv 1908.07257: pdf, ps.