Real spectrum of ring of definable functions

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.