## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### Real spectrum of ring of definable functions

Masato Fujita

#### 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.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 80, Number 6 (2004), 116-121.

Dates
First available in Project Euclid: 13 May 2005

https://projecteuclid.org/euclid.pja/1116014789

Digital Object Identifier
doi:10.3792/pjaa.80.116

Mathematical Reviews number (MathSciNet)
MR2075454

Zentralblatt MATH identifier
1059.03030

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality

#### Citation

Fujita, Masato. Real spectrum of ring of definable functions. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 6, 116--121. doi:10.3792/pjaa.80.116. https://projecteuclid.org/euclid.pja/1116014789

#### References

• 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).
• 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).
• Bochnak, J., Coste, M., and Roy, M.-F.: Real Algebraic Geometry. Results in Mathematics and Related Areas (3), vol. 36, Springer, Berlin (1998).
• Gamboa, J. M., and Ruiz, J. M.: On rings of abstract semialgebraic functions. Math. Z., 206, 527–532 (1991).
• Gillman, L., and Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960).
• Knight, J., Pillay, A., and Steinhorn, C.: Definable sets in ordered structures. II. Trans. Amer. Math. Soc., 295, 593–605 (1986).
• Miller, C.: Expansions of the real field with power functions. Ann. Pure Appl. Logic, 68, 79–94 (1994).
• Miller, C.: Exponention is hard to avoid. Proc. Amer. Math. Soc., 122, 257–259 (1994).
• van den Dries, L., and Miller, C.: Geometric categories and o-minimal structures. Duke Math. J., 84, 497–540 (1996).
• Pillay, A., and Steinhorn, C.: Definable sets in oredered structures. I. Trans. Amer. Math. Soc., 295, 565–592 (1986).
• 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).
• Dries, L. van den: Tame Topology and O-Minimal Structure. London Mathematical Society Lecture Note Series, 248, Cambridge Univ. Press, Cambridge (1998).