Skip to main content Accessibility help
×
×
Home

CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS IN p-OPTIMAL FIELDS

  • LUCK DARNIÈRE (a1) and IMMANUEL HALPUCZOK (a2)
  • Please note a correction has been issued for this article.

Abstract

We prove that for p-optimal fields (a very large subclass of p-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strong p-minimality. Then we turn to strongly p-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to a p-adic one. For such fields K, we prove that every definable subset of K × K d whose fibers over K are inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions on p-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

Copyright

References

Hide All
[1] Cubides-Kovacsics, P., Darnière, Luck, and Leenknegt, Eva, Topological Cell Decomposition and Dimension Theory in p-minimal Fields, preprint, 2015.
[2] Cubides-Kovacsics, Pablo and Leenknegt, Eva, Integration and cell decomposition in P-minimal structures, this Journal, to appear.
[3] Cluckers, Raf, Classification of semi-algebraic p-adic sets up to semi-algebraic bijection . Journal für die Reine und Angewandte Mathematik, vol. 540 (2001), pp. 105114.
[4] Cluckers, Raf, Presburger sets and P-minimal fields, this Journal, vol. 68 (2003), no. 1, pp. 153162.
[5] Cluckers, Raf, Analytic p-adic cell decomposition and integrals . Transactions of the American Mathematical Society, vol. 356 (2004), no. 4, pp. 14891499.
[6] Cluckers, Raf and Leenknegt, Eva, A version of p-adic minimality, this Journal, vol. 77 (2012), no. 2, pp. 621630.
[7] Denef, Jan, The rationality of the Poincaré series associated to the p-adic points on a variety . Inventiones Mathematicae, vol. 77 (1984), no. 1, pp. 123.
[8] Denef, Jan, p-adic semi-algebraic sets and cell decomposition . Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154166.
[9] Denef, Jan and Dries, Lou van den, p-adic and real subanalytic sets . Annals of Mathematics (2), vol. 128 (1988), no. 1, pp. 79138.
[10] Haskell, Deirdre and Macpherson, Dugald, A version of o-minimality for the p-adics. this Journal, vol. 62 (1997), no. 4, pp. 10751092.
[11] Macintyre, Angus. On definable subsets of p-adic fields, this Journal, vol. 41 (1976), no. 3, pp. 605610.
[12] Mourgues, Marie-Hélène, Corps p-minimaux avec fonctions de skolem définissables . Prépublications de l’équipe de logique de paris 7, Séminaire de structures algébriques ordonnées, 1999-2000.
[13] Mourgues, Marie-Hélène, Cell decomposition for P-minimal fields . Mathematical Logic Quarterly, vol. 55 (2009), no. 5, pp. 487492.
[14] Prestel, A. and Roquette, P., Formally p-adic Fields, Lecture Notes in Mathematics, vol. 1050, Springer-Verlag, 1984.
[15] Dries, Lou van den, Algebraic theories with definable Skolem functions, this Journal, vol. 49 (1984), no. 2, pp. 625629.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

A correction has been issued for this article: