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p-adically closed fields with nonstandard analytic structure

Published online by Cambridge University Press:  12 March 2014

Ali Bleybel
Ali Bleybel*
Affiliation:
Lebanese University, Faculty of Sciences Hadath, Lebanon. E-mail: bleybel@ul.edu.lb
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Abstract

We prove quantifier elimination for the field ℚp((tℚ)) (the completion of the field of Puiseux series over ℚp) in Macintyre's language together with symbols for functions in a class containing both t-adically and p-adically overconvergent functions. We also show that the theory of ℚp((tℚ)) is b-minimal in this language.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 802 - 816
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Bosch, S., Güntzer, U., and Remmert, R., Non-archimedean analysis, Springer-Verlag, 1984.CrossRefGoogle Scholar
[2]Clark, D. N., A note on the p-adic convergence of solutions of linear differential equations, Proceedings of the American Mathematical Society, vol. 17 (1966), pp. 262–269.Google Scholar
[3]Cluckers, R. and Lipshitz, L., Fields with analytic structure, preprint, 2007.Google Scholar
[4]Cluckers, R., Lipshitz, L., and Robinson, Z., Analytic cell decomposition and analytic motivic integration, Annales Scientifiques de l'École Normale Supérieure, vol. 39 (2006), no. 4, pp. 535–568.Google Scholar
[5]Cluckers, R., Lipshitz, L., and Robinson, Z., Real closedfields with standard and nonstandard analytic structures, Journal of the London Mathematical Society (2), vol. 78 (2008), no. 1, pp. 198–212.CrossRefGoogle Scholar
[6]Cluckers, R. and Loeser, F., b-minimality, Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 195–227.CrossRefGoogle Scholar
[7]Denef, J., The rationality of the Poincaré series associated to the p-adic points on a variety, Imentiones Mathematicae, vol. 77 (1984), pp. 1–23.CrossRefGoogle Scholar
[8]Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Annals of Mathematics, vol. 128 (1988), pp. 79–138.CrossRefGoogle Scholar
[9]Endler, O., Valuation theory, Springer-Verlag, 1972.CrossRefGoogle Scholar
[10]Fresnel, J. and van der Put, M., Géométrie analytique rigide et applications, Birkhäuser, Boston, 1981.Google Scholar
[11]Güntzer, U., Modellringe in der nichtarchimedischen Funktiontheorie, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. Series A, vol. 29 (1967), pp. 334–342.Google Scholar
[12]Lipshitz, L. and Robinson, Z., Rigid subanalytic subsets of curves and surfaces, Journal of the London Mathematical Society (2), vol. 59 (1999), pp. 895–921.CrossRefGoogle Scholar
[13]Lipshitz, L. and Robinson, Z., Overconvergent real closed quantifier elimination, Bulletin of the London Mathematical Societ, vol. 38 (2006), pp. 897–906.CrossRefGoogle Scholar
[14]Macintyre, A., On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605–610.Google Scholar
[15]Pas, J., Uniform p-adic cell decomposition and local zeta functions. Journal für die Reine und Angewandte Mathematik, vol. 399 (1989), pp. 137–172.Google Scholar
[16]Pas, J., Cell decomposition and zeta function in a tower of unramified extensions of a p-adic field, Proceedings of the London Mathematical Society, vol. 60 (1990), pp. 37–67.CrossRefGoogle Scholar
[17]Prestel, A. and Roquette, P., Formally p-adic fields, [B], Lecture Notes in Mathematics, 1050, Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
[18]Schikhof, W.H., Ultrametric calculus. An introduction to p-adic analysis, Cambridge Studies in Advanced Mathematics, no. 4, Cambridge University Press, 1984.Google Scholar
[19]Schoutens, H., Approximation and subanalytic sets over a complete valuation ring, Ph.D. thesis, Catholic University of Leuven, 1991.Google Scholar
[20]Scowcroft, P. and van den Dries, L., On the structure of semialgebraic sets over p-adic fields, this Journal, vol. 53 (1988), pp. 1138–1164.Google Scholar
[21]Setoyanaoi, M., Note on Clark's theorem for p-adic convergence, Proceedings of the American Mathematical Society, vol. 125 (1997), pp. 717–721.CrossRefGoogle Scholar
[22]Sibuya, Y. and Sperber, S., Arithmetic properties of power series solutions of algebraic differential equations, Annals of Mathematics, vol. 113 (1981), pp. 111–157.CrossRefGoogle Scholar
[23]van den Dries, L., Analytic Ax-Kochen-Ersov theorems, Contemporary Mathematics, vol. 131 (1992), pp. 379–398.CrossRefGoogle Scholar
[24]van den Dries, L., Haskell, D., and Macpherson, D., One-dimensional p-adic subanalytic sets, Journal of the London Mathematical Society (2), vol. 59 (1999), pp. 1–20.CrossRefGoogle Scholar