Hostname: page-component-7f64f4797f-kjzhn Total loading time: 0 Render date: 2025-11-03T09:38:07.614Z Has data issue: false hasContentIssue false
  • English
  • Français

DIFFERENTIATION IN P-MINIMAL STRUCTURES AND Ap-ADIC LOCAL MONOTONICITY THEOREM

Published online by Cambridge University Press:  12 December 2014

TRISTAN KUIJPERS  and
EVA LEENKNEGT
TRISTAN KUIJPERS
Affiliation:
KU LEUVEN DEPARTMENT OF MATHEMATICS CELESTIJNENLAAN 200B 3001 LEUVEN, BELGIUME-mail: tristan.kuijpers@wis.kuleuven.be
EVA LEENKNEGT
Affiliation:
PURDUE UNIVERSITY DEPARTMENT OF MATHEMATICS 150 N. UNIVERSITY STREET WEST LAFAYETTE, IN 47907-2067, USAE-mail: eleenkne@math.purdue.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a p-adic, local version of the Monotonicity Theorem forP-minimal structures. The existence of such a theorem wasoriginally conjectured by Haskell and Macpherson. We approach the problem byconsidering the first order strict derivative. In particular, we show that, fora wide class of P-minimal structures, the definable functionsf : K → K arealmost everywhere strictly differentiable and satisfy the Local JacobianProperty.

Keywords

P-minimalityp-adic differentiationmonotonicity

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 4 , December 2014 , pp. 1133 - 1147
Copyright
Copyright © The Association for Symbolic Logic 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Bertram, W., Glöckner, H., and Neeb, K.-H.. Differential calculus over general base fields and rings, Expositiones Mathematicae, vol. 22 (2004), no. 3, pp. 213–282.CrossRefGoogle Scholar
Cluckers, R., Comte, G., and Loeser, F., Lipschitz continuity properties for p-adic semi-algebraic and subanalytic functions, Geometric and Functional Analysis, vol. 20 (2010), no. 1, pp. 68–87.Google Scholar
Cluckers, R. and Halupczok, I., Approximations and lipschitz continuity in p-adic semi-algebraic and subanalytic geometry, Selecta Mathematica, vol. 18 (2012), no. 4, pp. 825–837.Google Scholar
Cluckers, R. and Lipshitz, L., Fields with analytic structure, Journal of the European Mathematical Society, vol. 13 (2011), no. 4, pp. 1147–1223.Google Scholar
Cluckers, Raf and Loeser, François, Constructible motivic functions and motivic integration, Inventiones mathematicae, vol. 173 (2008), no. 1, pp. 23–121.Google Scholar
Denef, J., The rationality of the Poincaré series associated to the p-adic points on a variety, Inventiones mathematicae, vol. 77 (1984), pp. 1–23.Google Scholar
Denef, J. and Dries, Lou van den, p-adic and real subanalytic sets, Annals of Mathematics,(2), vol. 128 (1988), no. 1, pp. 79–138.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
Hrushovski, Ehud and Kazhdan, David, Integration in valued fields, Algebraic geometry and number theory, Progress in Mathematics, vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 261–405.Google Scholar
Robert, Alain M., A course in p-adic analysis, Graduate Texts in Mathematics, vol. 198, Springer-Verlag, New York, 2000.Google Scholar
Schikhof, W. H., Ultrametric calculus:An introduction to p-adic analysis, Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, Cambridge, 1984.Google Scholar