## Nagoya Mathematical Journal

### Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation

Serge Randriambololona

#### Abstract

This article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.

#### Article information

Source
Nagoya Math. J., Volume 215 (2014), 225-237.

Dates
First available in Project Euclid: 14 July 2014

https://projecteuclid.org/euclid.nmj/1405342240

Digital Object Identifier
doi:10.1215/00277630-2781221

Mathematical Reviews number (MathSciNet)
MR3263529

Zentralblatt MATH identifier
1306.81217

#### Citation

Randriambololona, Serge. Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation. Nagoya Math. J. 215 (2014), 225--237. doi:10.1215/00277630-2781221. https://projecteuclid.org/euclid.nmj/1405342240

#### References

• [1] A. Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions, Invent. Math. 125 (1996), 1–12.
• [2] M. Karpinski and A. Macintyre, A generalization of Wilkie’s theorem of the complement, and an application to Pfaffian closure, Selecta Math. (N.S.) 5 (1999), 507–516.
• [3] F.-V. Kuhlmann and S. Kuhlmann, Valuation theory of exponential Hardy fields, I, Math. Z. 243 (2003), 671–688.
• [4] O. Le Gal, A generic condition implying o-minimality for restricted $C^{\infty}$-functions, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), 479–492.
• [5] C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), 79–94.
• [6] C. Miller, Exponentiation is hard to avoid, Proc. Amer. Math. Soc. 122 (1994), 257–259.
• [7] A. Nesin and A. Pillay, Some model theory of compact Lie groups, Trans. Amer. Math. Soc. 326 (1991), no. 1, 453–463.
• [8] B. Poizat, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, New York, 2000.
• [9] S. Randriambololona, o-Minimal structures: Low arity versus generation, Illinois J. Math. 49 (2005), 547–558.
• [10] R. Soufflet, Asymptotic expansions of logarithmic-exponential functions, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 125–146.
• [11] L. van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), 61–78.
• [12] L. van den Dries, Tame Topology and o-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge University Press, Cambridge, 1998.
• [13] L. van den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), 183–205.
• [14] L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994), 19–56.
• [15] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497–540.