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DIFFERENTIATION IN P-MINIMAL STRUCTURES AND A p-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: [email protected]
EVA LEENKNEGT
Affiliation:
PURDUE UNIVERSITY DEPARTMENT OF MATHEMATICS 150 N. UNIVERSITY STREET WEST LAFAYETTE, IN 47907-2067, USAE-mail: [email protected]
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Abstract

We prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions f : K → K are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.

Keywords

P-minimalityp-adic differentiationmonotonicity
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 

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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