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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.

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[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.
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[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.
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[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.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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