×

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.

MSC:

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

References:

[1] Andradas, C., and Becker, E.: A note on the real spectrum of analytic functions on an analytic manifold of dimension one. Real Analytic and Algebraic Geometry (Trento, 1988). Lecture Notes in Math., vol. 1420, Springer, Berlin-Heidelberg-New York, pp. 1-21 (1990). · Zbl 0779.32004 · doi:10.1007/BFb0083907
[2] Andradas, C., Bröcker, L., and Ruiz, J. M.: Constructible Sets in Real Geometry. Results in Mathematics and Related Areas (3), vol. 33, Springer, Berlin (1996). · Zbl 0873.14044
[3] Bochnak, J., Coste, M., and Roy, M.-F.: Real Algebraic Geometry. Results in Mathematics and Related Areas (3), vol. 36, Springer, Berlin (1998).
[4] Gamboa, J. M., and Ruiz, J. M.: On rings of abstract semialgebraic functions. Math. Z., 206 , 527-532 (1991). · Zbl 0745.14022 · doi:10.1007/BF02571359
[5] Gillman, L., and Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960). · Zbl 0093.30001
[6] Knight, J., Pillay, A., and Steinhorn, C.: Definable sets in ordered structures. II. Trans. Amer. Math. Soc., 295 , 593-605 (1986). · Zbl 0662.03024 · doi:10.2307/2000053
[7] Miller, C.: Expansions of the real field with power functions. Ann. Pure Appl. Logic, 68 , 79-94 (1994). · Zbl 0823.03018 · doi:10.1016/0168-0072(94)90048-5
[8] Miller, C.: Exponention is hard to avoid. Proc. Amer. Math. Soc., 122 , 257-259 (1994). · Zbl 0808.03022 · doi:10.2307/2160869
[9] van den Dries, L., and Miller, C.: Geometric categories and o-minimal structures. Duke Math. J., 84 , 497-540 (1996). · Zbl 0889.03025 · doi:10.1215/S0012-7094-96-08416-1
[10] Pillay, A., and Steinhorn, C.: Definable sets in oredered structures. I. Trans. Amer. Math. Soc., 295 , 565-592 (1986). · Zbl 0662.03023 · doi:10.2307/2000052
[11] Dries, L. van den: Remarks on Tarki’s problem concerning \((\textbf{R},+, \cdot, \exp)\). Logic Colloquium ’82. Stud. Logic Found. Math., vol. 112, North-Holland, Amsterdam, pp. 97-121 (1984). · Zbl 0585.03006 · doi:10.1016/S0049-237X(08)71811-1
[12] Dries, L. van den: Tame Topology and O-Minimal Structure. London Mathematical Society Lecture Note Series, 248, Cambridge Univ. Press, Cambridge (1998). · Zbl 0953.03045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.