Abstract
Consider an o-minimal expansion of the real field. We deal with the real spectrums of the ring $C_{\mathrm{df}}^r$ of definable $C^r$ functions on an affine definable $C^r$ manifold $M$ in the present paper. Here $r$ denotes a nonnegative integer. We show that the natural map $\operatorname{Sper}(C_{\mathrm{df}}^r(M)) \rightarrow \operatorname{Spec}(C_{\mathrm{df}}^r(M))$ 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 $\operatorname{Sper}(C_{\mathrm{df}}^r(M)) \rightarrow \operatorname{Spec}(C_{\mathrm{df}}^r(M))$ is a homeomorphism or not. However, the natural map $\operatorname{Sper}(C_{\mathrm{df}}^0(M)) \rightarrow \operatorname{Spec}(C_{\mathrm{df}}^0(M))$ is bijective even in this case.
Citation
Masato Fujita. "Real spectrum of ring of definable functions." Proc. Japan Acad. Ser. A Math. Sci. 80 (6) 116 - 121, June 2004. https://doi.org/10.3792/pjaa.80.116
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