Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Analytic $p$-adic cell decomposition and integrals

Author: Raf Cluckers
Journal: Trans. Amer. Math. Soc. 356 (2004), 1489-1499
MSC (2000): Primary 11S80, 32P05, 32B20; Secondary 03C10, 03C98, 11U09, 11S40
Published electronically: October 29, 2003
MathSciNet review: 2034315
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

References [Enhancements On Off] (What's this?)

  • 1. N. Bourbaki, Variétés différentielles et analytiques. Fascicule de résultats, Hermann, Paris, 1967, (French). MR 36:2161
  • 2. R. Cluckers, Classification of semialgebraic $p$-adic sets up to semialgebraic bijection, Journal für die reine und angewandte Mathematik 540 (2001), 105-114. MR 2002i:14052
  • 3. R. Cluckers and D. Haskell, Grothendieck rings of $\mathbb{Z} $-valued fields, Bulletin of Symbolic Logic 7 (2001), no. 2, 262-269. MR 2002g:03081
  • 4. J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Inventiones Mathematicae 77 (1984), 1-23. MR 86c:11043
  • 5. -, On the evaluation of certain $p$-adic integrals, Théorie des nombres, Sémin. Delange-Pisot-Poitou 1983-84, vol. 59, 1985, pp. 25-47.
  • 6. -, $p$-adic semialgebraic sets and cell decomposition, Journal für die reine und angewandte Mathematik 369 (1986), 154-166. MR 88d:11030
  • 7. -, Multiplicity of the poles of the Poincaré series of a $p$-adic subanalytic set, Sém. Th. Nombres Bordeaux 43 (1987-1988), 1-8. MR 90f:11089
  • 8. -, Arithmetic and geometric applications of quantifier elimination for valued fields, MSRI Publications, vol. 39, pp. 173-198, Cambridge University Press, 2000. MR 2001e:03063
  • 9. J. Denef and L. van den Dries, $p$-adic and real subanalytic sets, Annals of Mathematics 128 (1988), no. 1, 79-138. MR 89k:03034
  • 10. J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae 135 (1999), 201-232. MR 99k:14002
  • 11. -, Definable sets, motives and $p$-adic integrals, Journal of the American Mathematical Society 14 (2001), no. 2, 429-469. MR 2002k:14033
  • 12. L. van den Dries, Tame topology and o-minimal structures, Lecture note series, vol. 248, Cambridge University Press, 1998. MR 99j:03001
  • 13. L. van den Dries, D. Haskell, and D. Macpherson, One-dimensional $p$-adic subanalytic sets, Journal of the London Mathematical Society 59 (1999), no. 1, 1-20. MR 2000k:03077
  • 14. M.P.F. du Sautoy, Finitely generated groups, $p$-adic analytic groups and Poincaré series, Annals of Mathematics 137 (1993), no. 3, 639-670. MR 94j:20029
  • 15. W. Hodges, Model theory, Encyclopedia of Mathematics and Its Applications, vol. 42, Cambridge University Press, 1993. MR 94e:03002
  • 16. J. Igusa, Complex powers and asymptotic expansions I, Journal für die reine und angewandte Mathematik 268 (1974), 110-130. MR 50:254
  • 17. -, Complex powers and asymptotic expansions II, Journal für die reine und angewandte Mathematik 278 (1975), 307-321. MR 53:8018
  • 18. -, Lectures on forms of higher degree (notes by S. Raghavan), Lectures on mathematics and physics, Tata Institute of Fundamental Research, vol. 59, Springer-Verlag, 1978. MR 80m:10020
  • 19. B. Lichtin, On a question of Igusa: towards a theory of several variable asymptotic expansions I, Compositio Mathematica 120 (2000), no. 1, 25-82. MR 2001c:11086
  • 20. J.-M. Lion and J.-P. Rolin, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques (Integration of subanalytic functions and volumes of subanalytic subspaces), Ann. Inst. Fourier 48 (1998), no. 3, 755-767 (French). MR 2000i:32011
  • 21. L. Lipshitz and Z. Robinson, Rings of separated power series and quasi-affinoid geometry, Paris: Société Mathématique de France, vol. 264, Astérisque, 2000. MR 2001g:32017
  • 22. Nianzheng Liu, Analytic cell decomposition and the closure of $p$-adic semianalytic sets, Journal of Symbolic Logic 62 (1997), no. 1, 285-303. MR 98h:03039
  • 23. A. Macintyre, On definable subsets of $p$-adic fields, Journal of Symbolic Logic 41 (1976), 605-610. MR 58:5182
  • 24. M.-H. Mourgues, Corps p-minimaux avec fonctions de Skolem définissables, Séminaire de structures algébriques ordonnées, 1999-2000, prépublication de l'équipe de logique mathématique de Paris 7, pp. 1-8.
  • 25. A. Mylnikov, $p$-adic subanalytic preparation and cell decomposition theorems, Ph.D. thesis, Purdue University, 1999.
  • 26. J. Oesterlé, Réduction modulo $p^n$ des sous-ensembles analytiques fermés de $\mathbb{Z} _p^n$, Inventiones Mathematicae 66 (1982), no. 2, 325-341. MR 83j:12014
  • 27. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Publ. Math. Inst. Hautes Études Sci. 323-401 (1981). MR 83k:12011
  • 28. S. Wilcox, Topics in the model theory of $p$-adic numbers, Ph.D. thesis, University of Oxford, (unfinished).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11S80, 32P05, 32B20, 03C10, 03C98, 11U09, 11S40

Retrieve articles in all journals with MSC (2000): 11S80, 32P05, 32B20, 03C10, 03C98, 11U09, 11S40

Additional Information

Raf Cluckers
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Address at time of publication: École Normale Supérieure, Département de Mathématiques et Applications, 45 rue d’Ulm, 75230 Paris Cedex 05, France

Keywords: Subanalytic $p$-adic sets, cell decomposition, $p$-adic integrals, Igusa's local zeta functions
Received by editor(s): August 15, 2002
Published electronically: October 29, 2003
Additional Notes: The author is a Research Assistant of the Fund for Scientific Research – Flanders (Belgium)(F.W.O.)
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society