Analytic 𝑝-adic cell decomposition and integrals

Cluckers, Raf

doi:10.1090/S0002-9947-03-03458-5

Analytic $p$-adic cell decomposition and integrals
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Trans. Amer. Math. Soc. 356 (2004), 1489-1499 Request permission

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

N. Bourbaki, Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1333, Hermann, Paris, 1967 (French). MR 0219078
Raf Cluckers, Classification of semi-algebraic $p$-adic sets up to semi-algebraic bijection, J. Reine Angew. Math. 540 (2001), 105–114. MR 1868600, DOI 10.1515/crll.2001.081
Raf Cluckers and Deirdre Haskell, Grothendieck rings of $\Bbb Z$-valued fields, Bull. Symbolic Logic 7 (2001), no. 2, 262–269. MR 1839548
J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1–23. MR 751129, DOI 10.1007/BF01389133
—, On the evaluation of certain $p$-adic integrals, Théorie des nombres, Sémin. Delange-Pisot-Poitou 1983–84, vol. 59, 1985, pp. 25–47.
Jan Denef, $p$-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math. 369 (1986), 154–166. MR 850632, DOI 10.1515/crll.1986.369.154
J. Denef, Multiplicity of the poles of the Poincaré series of a $p$-adic subanalytic set, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988) Univ. Bordeaux I, Talence, [1988?], pp. Exp. No. 43, 8. MR 993136
Jan Denef, Arithmetic and geometric applications of quantifier elimination for valued fields, Model theory, algebra, and geometry, Math. Sci. Res. Inst. Publ., vol. 39, Cambridge Univ. Press, Cambridge, 2000, pp. 173–198. MR 1773707, DOI 10.2977/prims/1145476152
J. Denef and L. van den Dries, $p$-adic and real subanalytic sets, Ann. of Math. (2) 128 (1988), no. 1, 79–138. MR 951508, DOI 10.2307/1971463
Jan Denef and François Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), no. 1, 201–232. MR 1664700, DOI 10.1007/s002220050284
Jan Denef and François Loeser, Definable sets, motives and $p$-adic integrals, J. Amer. Math. Soc. 14 (2001), no. 2, 429–469. MR 1815218, DOI 10.1090/S0894-0347-00-00360-X
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
Lou van den Dries, Deirdre Haskell, and Dugald Macpherson, One-dimensional $p$-adic subanalytic sets, J. London Math. Soc. (2) 59 (1999), no. 1, 1–20. MR 1688485, DOI 10.1112/S0024610798006917
Marcus P. F. du Sautoy, Finitely generated groups, $p$-adic analytic groups and Poincaré series, Ann. of Math. (2) 137 (1993), no. 3, 639–670. MR 1217350, DOI 10.2307/2946534
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
Jun-ichi Igusa, Complex powers and asymptotic expansions. I. Functions of certain types, J. Reine Angew. Math. 268(269) (1974), 110–130. MR 347753, DOI 10.1515/crll.1974.268-269.110
Jun-ichi Igusa, Complex powers and asymptotic expansions. II. Asymptotic expansions, J. Reine Angew. Math. 278(279) (1975), 307–321. MR 404215, DOI 10.1515/crll.1975.278-279.307
Jun-ichi Igusa, Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 59, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546292
Ben Lichtin, On a question of Igusa: towards a theory of several variable asymptotic expansions. I, Compositio Math. 120 (2000), no. 1, 25–82. MR 1738214, DOI 10.1023/A:1001743909924
J.-M. Lion and J.-P. Rolin, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 755–767 (French, with English and French summaries). MR 1644093
Leonard Lipshitz and Zachary Robinson, Rings of separated power series and quasi-affinoid geometry, Astérisque 264 (2000), vi+171 (English, with English and French summaries). MR 1758887
Nianzheng Liu, Analytic cell decomposition and the closure of $p$-adic semianalytic sets, J. Symbolic Logic 62 (1997), no. 1, 285–303. MR 1450524, DOI 10.2307/2275742
Angus Macintyre, On definable subsets of $p$-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610. MR 485335, DOI 10.2307/2272038
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.
A. Mylnikov, $p$-adic subanalytic preparation and cell decomposition theorems, Ph.D. thesis, Purdue University, 1999.
Joseph Oesterlé, Réduction modulo $p^{n}$ des sous-ensembles analytiques fermés de $\textbf {Z}^{N}_{p}$, Invent. Math. 66 (1982), no. 2, 325–341 (French). MR 656627, DOI 10.1007/BF01389398
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
S. Wilcox, Topics in the model theory of $p$-adic numbers, Ph.D. thesis, University of Oxford, (unfinished).