Analytic cell decomposition and analytic motivic integration
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 39 (2006) no. 4, p. 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},
     publisher = {Elsevier},
     volume = {Ser. 4, 39},
     number = {4},
     year = {2006},
     pages = {535-568},
     doi = {10.1016/j.ansens.2006.03.001},
     zbl = {05125019},
     mrnumber = {2290137},
     language = {en},
     url = {http://www.numdam.org/item/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/item/ASENS_2006_4_39_4_535_0/

[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 201378 | MR 184930 | Zbl 0223.02050

[2] Bosch S., Güntzer U., Remmert R., Non-Archimedean Analysis, Springer, Berlin, 1984. | MR 746961 | Zbl 0539.14017

[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 2140048 | Zbl 1119.03028 | Zbl 02247429

[4] Cluckers R., Analytic p-adic cell decomposition and integrals, Trans. Amer. Math. Soc. 356 (2004) 1489-1499, math.NT/0206161. | MR 2034315 | Zbl 1048.11094

[5] Cluckers R., Lipshitz L., Fields with analytic structure, Preprint. | MR 2800487 | Zbl 1291.03074

[6] Cluckers R., Loeser F., Constructible motivic functions and motivic integration, math.AG/0410203. | Zbl 1179.14011

[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 2092754 | Zbl 1062.14030

[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 2099547 | Zbl 1064.14021

[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 2159379 | Zbl 1159.12314 | Zbl 05013095

[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 2188869 | Zbl 1081.14032

[11] Cohen P.J., Decision procedures for real and p-adic fields, Comm. Pure Appl. Math. 22 (1969) 131-151. | MR 244025 | Zbl 0167.01502

[12] Denef J., The rationality of the Poincaré series associated to the p-adic points on a variety, Inventiones Mathematicae 77 (1984) 1-23. | MR 751129 | Zbl 0537.12011

[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 902824 | Zbl 0597.12021

[14] Denef J., p-adic semialgebraic sets and cell decomposition, Journal für die reine und angewandte Mathematik 369 (1986) 154-166. | MR 850632 | Zbl 0584.12015

[15] Denef J., Loeser F., Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae 135 (1999) 201-232. | MR 1664700 | Zbl 0928.14004

[16] Denef J., Loeser F., Definable sets, motives and p-adic integrals, J. Amer. Math. Soc. 14 (2) (2001) 429-469. | MR 1815218 | Zbl 1040.14010

[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 2099079 | Zbl 1061.11067

[18] Denef J., Van Den Dries L., p-adic and real subanalytic sets, Ann. of Math. 128 (1988) 79-138. | MR 951508 | Zbl 0693.14012

[19] Van Den Dries L., Analytic Ax-Kochen-Ershov theorems, in: Contemporary Mathematics, vol. 131, 1992, pp. 379-398. | MR 1175894 | Zbl 0835.03004

[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 1688485 | Zbl 0932.03038

[22] Endler O., Valuation Theory, Springer, Berlin, 1972. | MR 357379 | Zbl 0257.12111

[23] Fresnel J., Van Der Put M., Géométrie analytique rigide et applications, Progr. Math., vol. 18, Birkhäuser, Basel, 1981. | MR 644799 | Zbl 0479.14015

[24] Fresnel J., Van Der Put M., Rigid Geometry and Applications, Progr. Math., vol. 218, Birkhäuser, Basel, 2004. | Zbl 1096.14014

[25] Kazhdan D., An algebraic integration, in: Mathematics: Frontiers and Perspectives, AMS, Providence, RI, 2000, pp. 93-115. | MR 1754770 | Zbl 0976.20030

[26] Kuhlmann F.-V., Quantifier elimination for Henselian fields relative to additive and multiplicative congruences, Israel J. Math. 85 (1994) 277-306. | MR 1264348 | Zbl 0809.03028

[27] Lang S., Algebra, Addison-Wesley, 1965. | MR 197234 | Zbl 0848.13001

[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 1644093 | Zbl 0912.32007

[29] Lipshitz L., Rigid subanalytic sets, Amer. J. Math. 115 (1993) 77-108. | MR 1209235 | Zbl 0792.14010

[30] Lipshitz L., Robinson Z., Rigid subanalytic subsets of the line and the plane, Amer. J. Math. 118 (1996) 493-527. | MR 1393258 | Zbl 0935.14035

[31] Lipshitz L., Robinson Z., Rigid subanalytic subsets of curves and surfaces, J. London Math. Soc. (2) 59 (1999) 895-921. | MR 1709087 | Zbl 0935.32008

[32] Lipshitz L., Robinson Z., Rings of separated power series, Astérisque 264 (2000) 3-108. | MR 1758887

[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 2156714 | Zbl 1081.03024

[35] Loeser F., Sebag J., Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J. 119 (2) (2003) 315-344. | MR 1997948 | Zbl 1078.14029

[36] Macintyre A., Rationality of p-adic Poincaré series: uniformity in p, Ann. Pure Appl. Logic 49 (1) (1990) 31-74. | MR 1076249 | Zbl 0731.12015

[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. | MR 1004136 | Zbl 0666.12014

[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 1023804 | Zbl 0659.12017

[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 2018563 | Zbl 1040.03031

[41] Sebag J., Rationalité des séries de Poincaré et des fonctions zêta motiviques, Manuscripta Math. 115 (2) (2004) 125-162, (in French). | MR 2098466 | Zbl 1073.14524

[42] Sebag J., Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132 (1) (2004) 1-54, (in French). | Numdam | MR 2075915 | Zbl 1084.14012