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TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

Published online by Cambridge University Press:  21 March 2017

PABLO CUBIDES KOVACSICS ,
LUCK DARNIÈRE  and
EVA LEENKNEGT
PABLO CUBIDES KOVACSICS
Affiliation:
LABORATOIRE NICOLAS ORESME UNIVERSITÉ DE CAEN CNRS U.M.R. 6139 F 14032 CAEN CEDEX, FRANCE E-mail: [email protected]
LUCK DARNIÈRE
Affiliation:
LAREMA, UNIVERSITÉ D’ANGERS 2 BD LAVOISIER, 49045 ANGERS CEDEX 01, FRANCE E-mail: [email protected]
EVA LEENKNEGT
Affiliation:
DEPARTMENT OF MATHEMATICS KULEUVEN CELESTIJNENLAAN 200B, 3001 HEVERLEE, BELGIUM E-mail: [email protected]
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Abstract

This paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.

In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

Keywords

P-minimalityp-adically closed fieldsdimensioncell decomposition
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 347 - 358
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Bochnak, J., Coste, M., and Roy, M.-F., Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 12, Springer-Verlag, Berlin, 1987.Google Scholar
Cluckers, R., Analytic p-adic cell decomposition and integrals . Transactions of the American Mathematical Society, vol. 356 (2004), no. 4, pp. 1489–1499.Google Scholar
Cluckers, R. and Loeser, F., b-minimality . Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 195–227.CrossRefGoogle Scholar
Cohen, P. J., Decision procedures for real and p-adic fields . Communications on Pure and Applied Mathematics, vol. 22 (1969), pp. 131–151.Google Scholar
Cubides Kovacsics, P. and Leenknegt, E., Integration and cell decomposition in P-minimal structures, this Journal, vol. 81 (2016), pp. 1124–1141.Google Scholar
Cubides Kovacsics, P. and Nguyen, K. H., A P-minimal field without definable skolem functions, this Journal, to appear, arXiv:1605.00945.Google Scholar
Darnière, L. and Halupczok, I., Cell decomposition and dimension theory in p-optimal fields, this Journal, vol. 82 (2017), no. 1, pp. 120–136.Google Scholar
Denef, J., The rationality of the Poincaré series associated to the p-adic points on a variety . Inventiones Mathematicae, vol. 77 (1984), no. 1, pp. 1–23.Google Scholar
Denef, J., p-adic semi-algebraic sets and cell decomposition . Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154–166.Google Scholar
van den Dries, L., Dimension of definable sets, algebraic boundedness and Henselian fields . Annals of Pure and Applied Logic, vol. 45 (1989), no. 2, pp. 189–209, Stability in model theory, II (Trento, 1987).Google Scholar
van den Dries, L., Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.Google Scholar
Haskell, D. and Macpherson, D., A version of o-minimality for the p-adics, this Journal, vol. 62 (1997), no. 4, pp. 1075–1092.Google Scholar
Kuijpers, T. and Leenknegt, E., Differentiation in P-minimal structures and a p-adic local monotonicity theorem, this Journal, vol. 79 (2014), no. 4, pp. 1133–1147.Google Scholar
Mathews, L., Cell decomposition and dimension functions in first-order topological structures . Proceedings of the London Mathematical Society, Third Series, vol. 70 (1995), no. 1, pp. 1–32.Google Scholar
Mourgues, M-H., Cell decomposition for P-minimal fields . Mathematical Logic Quarterly, vol. 55 (2009), no. 5, pp. 487–492.Google Scholar
Pas, J., Cell decomposition and local zeta functions in a tower of unramified extensions of a p-adic field . Proceedings of the London Mathematical Society, Third Series, vol. 60 (1990), no. 1, pp. 37–67.Google Scholar