Skip to main content Accessibility help
×
Home

TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

  • PABLO CUBIDES KOVACSICS (a1), LUCK DARNIÈRE (a2) and EVA LEENKNEGT (a3)

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.

Copyright

References

Hide All
[1] Bochnak, J., Coste, M., and Roy, M.-F., Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 12, Springer-Verlag, Berlin, 1987.
[2] Cluckers, R., Analytic p-adic cell decomposition and integrals . Transactions of the American Mathematical Society, vol. 356 (2004), no. 4, pp. 14891499.
[3] Cluckers, R. and Loeser, F., b-minimality . Journal of Mathematical Logic, vol. 7 (2007), no. 2, pp. 195227.
[4] Cohen, P. J., Decision procedures for real and p-adic fields . Communications on Pure and Applied Mathematics, vol. 22 (1969), pp. 131151.
[5] Cubides Kovacsics, P. and Leenknegt, E., Integration and cell decomposition in P-minimal structures, this Journal, vol. 81 (2016), pp. 11241141.
[6] Cubides Kovacsics, P. and Nguyen, K. H., A P-minimal field without definable skolem functions, this Journal, to appear, arXiv:1605.00945.
[7] Darnière, L. and Halupczok, I., Cell decomposition and dimension theory in p-optimal fields, this Journal, vol. 82 (2017), no. 1, pp. 120136.
[8] 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. 123.
[9] Denef, J., p-adic semi-algebraic sets and cell decomposition . Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154166.
[10] 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. 189209, Stability in model theory, II (Trento, 1987).
[11] van den Dries, L., Tame Topology and O-minimal Structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.
[12] Haskell, D. and Macpherson, D., A version of o-minimality for the p-adics, this Journal, vol. 62 (1997), no. 4, pp. 10751092.
[13] 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. 11331147.
[14] 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. 132.
[15] Mourgues, M-H., Cell decomposition for P-minimal fields . Mathematical Logic Quarterly, vol. 55 (2009), no. 5, pp. 487492.
[16] 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. 3767.

Keywords

TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

  • PABLO CUBIDES KOVACSICS (a1), LUCK DARNIÈRE (a2) and EVA LEENKNEGT (a3)

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