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

Published online by Cambridge University Press:  24 January 2017

LUCK DARNIÈRE
Affiliation:
FACULTÉ DES SCIENCES 2 BOULEVARD LAVOISIER 49045 ANGERS CEDEX 01, FRANCE
IMMANUEL HALPUCZOK
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS WOODHOUS LANE, LEEDS, UK

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 × Kd 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.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Cubides-Kovacsics, P., Darnière, Luck, and Leenknegt, Eva, Topological Cell Decomposition and Dimension Theory in p-minimal Fields, preprint, 2015.Google Scholar
Cubides-Kovacsics, Pablo and Leenknegt, Eva, Integration and cell decomposition in P-minimal structures, this Journal, to appear.Google Scholar
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.Google Scholar
Cluckers, Raf, Presburger sets and P-minimal fields, this Journal, vol. 68 (2003), no. 1, pp. 153162.Google Scholar
Cluckers, Raf, Analytic p-adic cell decomposition and integrals . Transactions of the American Mathematical Society, vol. 356 (2004), no. 4, pp. 14891499.CrossRefGoogle Scholar
Cluckers, Raf and Leenknegt, Eva, A version of p-adic minimality, this Journal, vol. 77 (2012), no. 2, pp. 621630.Google Scholar
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.CrossRefGoogle Scholar
Denef, Jan, p-adic semi-algebraic sets and cell decomposition . Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154166.Google Scholar
Denef, Jan and Dries, Lou van den, p-adic and real subanalytic sets . Annals of Mathematics (2), vol. 128 (1988), no. 1, pp. 79138.CrossRefGoogle Scholar
Haskell, Deirdre and Macpherson, Dugald, A version of o-minimality for the p-adics. this Journal, vol. 62 (1997), no. 4, pp. 10751092.Google Scholar
Macintyre, Angus. On definable subsets of p-adic fields, this Journal, vol. 41 (1976), no. 3, pp. 605610.Google Scholar
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.Google Scholar
Mourgues, Marie-Hélène, Cell decomposition for P-minimal fields . Mathematical Logic Quarterly, vol. 55 (2009), no. 5, pp. 487492.CrossRefGoogle Scholar
Prestel, A. and Roquette, P., Formally p-adic Fields, Lecture Notes in Mathematics, vol. 1050, Springer-Verlag, 1984.CrossRefGoogle Scholar
Dries, Lou van den, Algebraic theories with definable Skolem functions, this Journal, vol. 49 (1984), no. 2, pp. 625629.Google Scholar