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Unexpected imaginaries in valued fields with analytic structure

Published online by Cambridge University Press:  12 March 2014

Deirdre Haskell ,
Ehud Hrushovski  and
Dugald Macpherson
Deirdre Haskell
Affiliation:
Department of Mathematics and Statistics, Mcmaster University, 1280 Main ST W., Hamilton On L8S 4K1, Canada, E-mail:[email protected]
Ehud Hrushovski
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel, E-mail:[email protected]
Dugald Macpherson
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK, E-mail:[email protected]
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Abstract

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries in the corresponding algebraic setting.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 523 - 542
Copyright
Copyright © Association for Symbolic Logic 2013

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References

REFERENCES

[1] Baisalov, Y. and Poizat, B., Paires de structures o-minimales, this Journal, vol. 63 (1998), pp. 570–578.Google Scholar
[2] Celikler, Y.F., Dimension theory and parametrised normalization for d-semianalytic sets over nonarchimedean fields, this Journal, vol. 70 (2005), pp. 593–618.Google Scholar
[3] Cherlin, G. and Dickmann, M. A., Real closed rings II: model theory, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 213–231.CrossRefGoogle Scholar
[4] Cluckers, R., Presburger sets and p-minimal fields, this Journal, vol. 68 (2003), pp. 153–162.Google Scholar
[5] Cluckers, R. and Lipshitz, L., Fields with analytic structure, Journal of the European Mathematical Society, vol. 13 (2011), pp. 1147–1223.CrossRefGoogle Scholar
[6] Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Annals of Mathematics, vol. 128 (1988), pp. 79–138.CrossRefGoogle Scholar
[7] Dickmann, M.A., Elimination of quantifiers for ordered valuation rings, this Journal, vol. 52 (1987), pp. 116–128.Google Scholar
[8] van den Dries, L., t-convexity and tame extensions II, this Journal, vol. 62 (1997)), pp. 14–34.Google Scholar
[9] van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.CrossRefGoogle Scholar
[10] van den Dries, L., Haskell, D., and Macpherson, H.D., One-dimensional p-adic subanalytic sets, Journal of the London Mathematical Society. Second Series, vol. 59 (1999), pp. 1–20.CrossRefGoogle Scholar
[11] van den Dries, L. and Lewenberg, A.. t-convexity and tame extensions, this Journal, vol. 60 (1995), pp. 74–102.Google Scholar
[12] Haskell, D., Hrushovski, E., and Macpherson, H. D., Definable sets in algebraically closed valued fields: elimination of imaginaries. Journal fur die reine und angewandte Mathematik, vol. 597 (2006), pp. 175–236.Google Scholar
[13] Haskell, D. and Macpherson, H.D., Cell decompositions of c-minimal structures. Annals of Pure and Applied Logic, vol. 66 (1994), pp. 113–162.CrossRefGoogle Scholar
[14] Haskell, D., A version of o-minimality for the p-adics, this Journal, vol. 62 (1997). pp. 1075–1092.Google Scholar
[15] Holly, J.E., Canonical forms for definable subsets of algebraically closed and real closed valued fields, this Journal, vol. 60 (1995), pp. 843–860.Google Scholar
[16] Hrushovski, E. and Kazhdan, D., Integration in valued fields. Algebraic geometry and number theory: in honour of Vladimir Drinfeld's 50th birthday (Ginzburg, V., editor). Progress in Mathematics, vol. 253, Birkhauser, 2006, pp. 261–405.CrossRefGoogle Scholar
[17] Hrushovski, E. and Martin, B., Zeta functions from definable equivalence relations, arXiv:math/0701011.Google Scholar
[18] Lipshitz, L., Rigid subanalytic sets, American Journal of Mathematics, vol. 115 (1993). pp. 77–108.CrossRefGoogle Scholar
[19] Lipshitz, L. and Robinson, Z.. One-dimensional fibers of rigid subanalytic sets, this Journal, vol. 63 (1998), pp. 83–88.Google Scholar
[20] Macpherson, H.D. and Steinhorn, C., On variants of o-minimality. Annals of Pure and Applied Logic, vol. 79 (1996), pp. 165–209.CrossRefGoogle Scholar
[21] Mellor, T., Imaginaries in real closed valued fields, Annals of Pure and Applied Logic, vol. 139 (2006), pp. 230–279.CrossRefGoogle Scholar
[22] Miller, C., A growth dichotomy for o-minimal expansions oforderedfields. Logic: from foundations to applications (European Logic Colloquium 1993) (Hodges, W.. Hyland, J.M.. Steinhorn, C., and Truss, J.K., editors), Oxford University Press, 1996, pp. 385–399.CrossRefGoogle Scholar
[23] Poizat, B., Groupes stables, Nur al-Mantiq wal-Ma'rifah, Villeurbanne, 1987 (in English, Stable groups, American Mathematical Society, Providence, Rhode Island, 2001).Google Scholar
[24] du Sautoy, M., Finitely generated groups, p-adic analytic groups and Poincare scries. Annals of Mathematics. Second Series, vol. 137 (1993). no. 3. pp. 639–670.CrossRefGoogle Scholar