Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-04-27T11:03:57.420Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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