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On the decidability of the real field with a generic power function

Published online by Cambridge University Press:  12 March 2014

Gareth Jones  and
Tamara Servi
Gareth Jones
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK, E-mail: [email protected]
Tamara Servi
Affiliation:
Centro de Matemática e Aplicações Fundamentais, Av. Prof. Gama Pinto 2,1649-003 Lisboa, Portugal, E-mail: [email protected]
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Abstract

We show that the theory of the real field with a generic real power function is decidable, relative to an oracle for the rational cut of the exponent of the power function. We also show the existence of generic computable real numbers, hence providing an example of a decidable o-minimal proper expansion of the real field by an analytic function.

Keywords

real power functionsdecidabilityo-minimalityPrimary 03C6403B25
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1418 - 1428
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[BKW08]Bays, M., Kirby, J., and Wilkie, A., A Schanuel property for exponentially transcendental powers, The Bulletin of the London Mathematical Society, (to appear).Google Scholar
[Dries85]den Dries, L. Van, The field of reals with a predicate for the powers of two, Manuscripta Mathematica, vol. 54 (1985), no. 1–2, pp. 187–195.CrossRefGoogle Scholar
[Dries86]den Dries, L. Van, A generalization of the Tarski–Seidenberg theorem, and some nondefinability results, American Mathematical Society. Bulletin, vol. 15 (1986), no. 2.Google Scholar
[FS10]Fornasiero, A. and Servi, T., Definably complete Baire structures, Fundamenta Mathematicae, vol. 209 (2010), no. 3, pp. 215–241.CrossRefGoogle Scholar
[JW08]Jones, G. and Wilkie, A., Locally polynomially bounded structures, The Bulletin of the London Mathematical Society, vol. 40 (2008), pp. 239–248.CrossRefGoogle Scholar
[Khovanskii91]Khovanskii, A. G., Fewnomials, Translations of Mathematical Monographs, vol. 88, American Mathematical Society, Providence, RI, 1991.CrossRefGoogle Scholar
[Lang66]Lang, S., Algebra, Addison-Wesley Publishing Co., Reading, MA, 1966.Google Scholar
[MW96]Macintyre, A. and Wilkie, A., On the decidability of the real exponential field, Kreiseliana (Odifreddi, P., editor), A. K. Peters, Wellesley, MA, 1996, pp. 441–467.Google Scholar
[Miller-C94]Miller, C., Expansions of the realfield with power functions, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 79–94.CrossRefGoogle Scholar
[Miller-D10]Miller, D., Constructing o-minimal structures with decidable theories using generic families offunctions from quasianalytic classes.Google Scholar
[Servi08]Servi, T., On the first-order theory of real exponentiation, Edizioni della Normale, Pisa, 2008.Google Scholar
[Tarski51]Tarski, A., A decision method for elementary algebra and geometry, 2nd ed., University of California Press, Berkeley and Los Angeles, California, 1951.CrossRefGoogle Scholar
[WaldschmidtOO]Waldschmidt, M., Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften 326, Springer-Verlag, Berlin, 2000.CrossRefGoogle Scholar
[Wilkie96]Wilkie, A. J., Model-completeness results for expansions of the orderedfield of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, vol. 9 (1996), pp. 1051–1094.CrossRefGoogle Scholar