Real spectrum of ring of definable functions. (English) Zbl 1059.03030

Summary: Consider an o-minimal expansion of the real field. We deal with the real spectra of the ring \(C^r_{\text{df}}(M)\) of definable \(C^r\) functions on an affine definable \(C^r\) manifold \(M\). Here \(r\) denotes a nonnegative integer. We show that the natural map \(\text{Sper}(C^r_{\text{df}}(M))\to\text{Spec}(C^r_{\text{df}}(M))\) is a homeomorphism when the o-minimal structure is polynomially bounded. If the o-minimal structure is not polynomially bounded, it is not known whether the natural map \(\text{Sper}(C^r_{\text{df}} (M))\to\text{Spec}(C^r_{\text{df}}(M))\) is a homeomorphism or not. However, the natural map \(\text{Sper}(C^0_{\text{df}} (M))\to \text{Spec}(C^0_{\text{df}}(M))\) is bijective even in this case.


03C64 Model theory of ordered structures; o-minimality
13J30 Real algebra
Full Text: DOI Euclid


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