The prime and maximal spectra of rings of C^r semialgebraic functions are homeomorphic to those of the ring of continuous semialgebraic functions, which is their real closure.
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Rings of differentiable semialgebraic functions
The prime and maximal spectra of rings of C^r semialgebraic functions are homeomorphic to those of the ring of continuous semialgebraic functions, which is their real closure.