Analytic cell decomposition and analytic motivic integration
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 39 (2006) no. 4, pp. 535-568.
@article{ASENS_2006_4_39_4_535_0,
     author = {Cluckers, Raf and Lipshitz, Leonard and Robinson, Zachary},
     title = {Analytic cell decomposition and analytic motivic integration},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {535--568},
     publisher = {Elsevier},
     volume = {Ser. 4, 39},
     number = {4},
     year = {2006},
     doi = {10.1016/j.ansens.2006.03.001},
     mrnumber = {2290137},
     zbl = {05125019},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.ansens.2006.03.001/}
}
TY  - JOUR
AU  - Cluckers, Raf
AU  - Lipshitz, Leonard
AU  - Robinson, Zachary
TI  - Analytic cell decomposition and analytic motivic integration
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2006
SP  - 535
EP  - 568
VL  - 39
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.ansens.2006.03.001/
DO  - 10.1016/j.ansens.2006.03.001
LA  - en
ID  - ASENS_2006_4_39_4_535_0
ER  - 
%0 Journal Article
%A Cluckers, Raf
%A Lipshitz, Leonard
%A Robinson, Zachary
%T Analytic cell decomposition and analytic motivic integration
%J Annales scientifiques de l'École Normale Supérieure
%D 2006
%P 535-568
%V 39
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.ansens.2006.03.001/
%R 10.1016/j.ansens.2006.03.001
%G en
%F ASENS_2006_4_39_4_535_0
Cluckers, Raf; Lipshitz, Leonard; Robinson, Zachary. Analytic cell decomposition and analytic motivic integration. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 39 (2006) no. 4, pp. 535-568. doi : 10.1016/j.ansens.2006.03.001. http://www.numdam.org/articles/10.1016/j.ansens.2006.03.001/

[1] Ax J., Kochen S., Diophantine problems over local fields I, II, Amer. J. Math. 87 (1965) 605-648, Diophantine problems over local fields III, Ann. of Math. 83 (1966) 437-456. | MR | Zbl

[2] Bosch S., Güntzer U., Remmert R., Non-Archimedean Analysis, Springer, Berlin, 1984. | MR | Zbl

[3] Çelikler Y.F., Dimension theory and parameterized normalization for D-semianalytic sets over non-Archimedean fields, J. Symbolic Logic 70 (2005) 593-618. | MR | Zbl

[4] Cluckers R., Analytic p-adic cell decomposition and integrals, Trans. Amer. Math. Soc. 356 (2004) 1489-1499, math.NT/0206161. | MR | Zbl

[5] Cluckers R., Lipshitz L., Fields with analytic structure, Preprint. | MR | Zbl

[6] Cluckers R., Loeser F., Constructible motivic functions and motivic integration, math.AG/0410203. | Zbl

[7] Cluckers R., Loeser F., Fonctions constructibles et intégration motivique I, Comptes rendus de l'Académie des Sciences 339 (2004) 411-416, math.AG/0403349. | MR | Zbl

[8] Cluckers R., Loeser F., Fonctions constructibles et intégration motivique II, Comptes rendus de l'Académie des Sciences 339 (2004) 487-492, math.AG/0403350. | MR | Zbl

[9] Cluckers R., Loeser F., Ax-Kochen-Eršov theorems for p-adic integrals and motivic integration, in: Geometric Methods in Algebra and Number Theory, Progr. Math., vol. 235, Birkhäuser Boston, 2005, pp. 109-137. | MR | Zbl

[10] Cluckers R., Loeser F., Fonctions constructibles exponentielles, transformation de Fourier motivique et principe de transfert, Comptes rendus de l'Académie des Sciences 341 (2005) 741-746, math.NT/0509723. | MR | Zbl

[11] Cohen P.J., Decision procedures for real and p-adic fields, Comm. Pure Appl. Math. 22 (1969) 131-151. | MR | Zbl

[12] Denef J., The rationality of the Poincaré series associated to the p-adic points on a variety, Inventiones Mathematicae 77 (1984) 1-23. | EuDML | MR | Zbl

[13] Denef J., On the evaluation of certain p-adic integrals, in: Séminaire de théorie des nombres, Paris 1983-84, Progr. Math., vol. 59, Birkhäuser Boston, 1985, pp. 25-47. | MR | Zbl

[14] Denef J., p-adic semialgebraic sets and cell decomposition, Journal für die reine und angewandte Mathematik 369 (1986) 154-166. | EuDML | MR | Zbl

[15] Denef J., Loeser F., Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae 135 (1999) 201-232. | MR | Zbl

[16] Denef J., Loeser F., Definable sets, motives and p-adic integrals, J. Amer. Math. Soc. 14 (2) (2001) 429-469. | MR | Zbl

[17] Denef J., Loeser F., On some rational generating series occurring in arithmetic geometry, in: Adolphson A., Baldassarri F., Berthelot P., Katz N., Loeser F. (Eds.), Geometric Aspects of Dwork Theory 1, de Gruyter, Berlin, 2004, pp. 509-526, math.NT/0212202. | MR | Zbl

[18] Denef J., Van Den Dries L., p-adic and real subanalytic sets, Ann. of Math. 128 (1988) 79-138. | MR | Zbl

[19] Van Den Dries L., Analytic Ax-Kochen-Ershov theorems, in: Contemporary Mathematics, vol. 131, 1992, pp. 379-398. | MR | Zbl

[20] van den Dries L., notes on cell decomposition.

[21] Van Den Dries L., Haskell D., Macpherson D., One-dimensional p-adic subanalytic sets, J. London Math. Soc. (2) 59 (1999) 1-20. | MR | Zbl

[22] Endler O., Valuation Theory, Springer, Berlin, 1972. | MR | Zbl

[23] Fresnel J., Van Der Put M., Géométrie analytique rigide et applications, Progr. Math., vol. 18, Birkhäuser, Basel, 1981. | MR | Zbl

[24] Fresnel J., Van Der Put M., Rigid Geometry and Applications, Progr. Math., vol. 218, Birkhäuser, Basel, 2004. | Zbl

[25] Kazhdan D., An algebraic integration, in: Mathematics: Frontiers and Perspectives, AMS, Providence, RI, 2000, pp. 93-115. | MR | Zbl

[26] Kuhlmann F.-V., Quantifier elimination for Henselian fields relative to additive and multiplicative congruences, Israel J. Math. 85 (1994) 277-306. | MR | Zbl

[27] Lang S., Algebra, Addison-Wesley, 1965. | MR | Zbl

[28] Lion J.-M., Rolin J.-P., Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier 48 (3) (1998) 755-767, (in French). | Numdam | MR | Zbl

[29] Lipshitz L., Rigid subanalytic sets, Amer. J. Math. 115 (1993) 77-108. | MR | Zbl

[30] Lipshitz L., Robinson Z., Rigid subanalytic subsets of the line and the plane, Amer. J. Math. 118 (1996) 493-527. | MR | Zbl

[31] Lipshitz L., Robinson Z., Rigid subanalytic subsets of curves and surfaces, J. London Math. Soc. (2) 59 (1999) 895-921. | MR | Zbl

[32] Lipshitz L., Robinson Z., Rings of separated power series, Astérisque 264 (2000) 3-108. | Numdam | MR

[33] Lipshitz L., Robinson Z., Model completeness and subanalytic sets, Astérisque 264 (2000) 109-126.

[34] Lipshitz L., Robinson Z., Uniform properties of rigid subanalytic sets, Trans. Amer. Math. Soc. 357 (1) (2005) 4349-4377. | MR | Zbl

[35] Loeser F., Sebag J., Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J. 119 (2) (2003) 315-344. | MR | Zbl

[36] Macintyre A., Rationality of p-adic Poincaré series: uniformity in p, Ann. Pure Appl. Logic 49 (1) (1990) 31-74. | MR | Zbl

[37] Nicaise J., Sebag J., Invariant de Serre et fibre de Milnor analytique, Available at:, http://www.wis.kuleuven.ac.be/algebra/artikels/artikelse.htm, (in French).

[38] Pas J., Uniform p-adic cell decomposition and local zeta-functions, Journal für die reine und angewandte Mathematik 399 (1989) 137-172. | EuDML | MR | Zbl

[39] Pas J., Cell decomposition and local zeta-functions in a tower of unramified extensions of a p-adic field, Proc. London Math. Soc. 60 (1990) 37-67. | MR | Zbl

[40] Scanlon T., Quantifier elimination for the relative Frobenius, in: Kuhlmann Franz-Viktor, Kuhlmann Salma, Marshall Murray (Eds.), Valuation Theory and Its Applications, vol. II, Conference Proceedings of the International Conference on Valuation Theory (Saskatoon, 1999), Fields Institute Communications Series, AMS, Providence, RI, 2003, pp. 323-352. | MR | Zbl

[41] Sebag J., Rationalité des séries de Poincaré et des fonctions zêta motiviques, Manuscripta Math. 115 (2) (2004) 125-162, (in French). | MR | Zbl

[42] Sebag J., Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132 (1) (2004) 1-54, (in French). | EuDML | Numdam | MR | Zbl

Cited by Sources: