Analytic cell decomposition and analytic motivic integration

Raf Cluckers; Leonard Lipshitz; Zachary Robinson

Annales scientifiques de l'École Normale Supérieure (2006)

  • Volume: 39, Issue: 4, page 535-568
  • ISSN: 0012-9593

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Cluckers, Raf, Lipshitz, Leonard, and Robinson, Zachary. "Analytic cell decomposition and analytic motivic integration." Annales scientifiques de l'École Normale Supérieure 39.4 (2006): 535-568. <http://eudml.org/doc/82694>.

@article{Cluckers2006,
author = {Cluckers, Raf, Lipshitz, Leonard, Robinson, Zachary},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Henselian valued fields; cell decomposition; analytic structure; motivic integration},
language = {eng},
number = {4},
pages = {535-568},
publisher = {Elsevier},
title = {Analytic cell decomposition and analytic motivic integration},
url = {http://eudml.org/doc/82694},
volume = {39},
year = {2006},
}

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
PB - Elsevier
VL - 39
IS - 4
SP - 535
EP - 568
LA - eng
KW - Henselian valued fields; cell decomposition; analytic structure; motivic integration
UR - http://eudml.org/doc/82694
ER -

References

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  1. [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. Zbl0223.02050MR184930
  2. [2] Bosch S., Güntzer U., Remmert R., Non-Archimedean Analysis, Springer, Berlin, 1984. Zbl0539.14017MR746961
  3. [3] Çelikler Y.F., Dimension theory and parameterized normalization for D-semianalytic sets over non-Archimedean fields, J. Symbolic Logic70 (2005) 593-618. Zbl1119.03028MR2140048
  4. [4] Cluckers R., Analytic p-adic cell decomposition and integrals, Trans. Amer. Math. Soc.356 (2004) 1489-1499, math.NT/0206161. Zbl1048.11094MR2034315
  5. [5] Cluckers R., Lipshitz L., Fields with analytic structure, Preprint. Zbl1291.03074
  6. [6] Cluckers R., Loeser F., Constructible motivic functions and motivic integration, math.AG/0410203. Zbl1179.14011
  7. [7] Cluckers R., Loeser F., Fonctions constructibles et intégration motivique I, Comptes rendus de l'Académie des Sciences339 (2004) 411-416, math.AG/0403349. Zbl1062.14030MR2092754
  8. [8] Cluckers R., Loeser F., Fonctions constructibles et intégration motivique II, Comptes rendus de l'Académie des Sciences339 (2004) 487-492, math.AG/0403350. Zbl1064.14021MR2099547
  9. [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. Zbl1159.12314MR2159379
  10. [10] Cluckers R., Loeser F., Fonctions constructibles exponentielles, transformation de Fourier motivique et principe de transfert, Comptes rendus de l'Académie des Sciences341 (2005) 741-746, math.NT/0509723. Zbl1081.14032MR2188869
  11. [11] Cohen P.J., Decision procedures for real and p-adic fields, Comm. Pure Appl. Math.22 (1969) 131-151. Zbl0167.01502MR244025
  12. [12] Denef J., The rationality of the Poincaré series associated to the p-adic points on a variety, Inventiones Mathematicae77 (1984) 1-23. Zbl0537.12011MR751129
  13. [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. Zbl0597.12021MR902824
  14. [14] Denef J., p-adic semialgebraic sets and cell decomposition, Journal für die reine und angewandte Mathematik369 (1986) 154-166. Zbl0584.12015MR850632
  15. [15] Denef J., Loeser F., Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae135 (1999) 201-232. Zbl0928.14004MR1664700
  16. [16] Denef J., Loeser F., Definable sets, motives and p-adic integrals, J. Amer. Math. Soc.14 (2) (2001) 429-469. Zbl1040.14010MR1815218
  17. [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. Zbl1061.11067MR2099079
  18. [18] Denef J., van den Dries L., p-adic and real subanalytic sets, Ann. of Math.128 (1988) 79-138. Zbl0693.14012MR951508
  19. [19] van den Dries L., Analytic Ax–Kochen–Ershov theorems, in: Contemporary Mathematics, vol. 131, 1992, pp. 379-398. Zbl0835.03004MR1175894
  20. [20] van den Dries L., notes on cell decomposition. 
  21. [21] van den Dries L., Haskell D., Macpherson D., One-dimensional p-adic subanalytic sets, J. London Math. Soc. (2)59 (1999) 1-20. Zbl0932.03038MR1688485
  22. [22] Endler O., Valuation Theory, Springer, Berlin, 1972. Zbl0257.12111MR357379
  23. [23] Fresnel J., van der Put M., Géométrie analytique rigide et applications, Progr. Math., vol. 18, Birkhäuser, Basel, 1981. Zbl0479.14015MR644799
  24. [24] Fresnel J., van der Put M., Rigid Geometry and Applications, Progr. Math., vol. 218, Birkhäuser, Basel, 2004. Zbl1096.14014
  25. [25] Kazhdan D., An algebraic integration, in: Mathematics: Frontiers and Perspectives, AMS, Providence, RI, 2000, pp. 93-115. Zbl0976.20030MR1754770
  26. [26] Kuhlmann F.-V., Quantifier elimination for Henselian fields relative to additive and multiplicative congruences, Israel J. Math.85 (1994) 277-306. Zbl0809.03028MR1264348
  27. [27] Lang S., Algebra, Addison-Wesley, 1965. Zbl0848.13001MR197234
  28. [28] Lion J.-M., Rolin J.-P., Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier48 (3) (1998) 755-767, (in French). Zbl0912.32007MR1644093
  29. [29] Lipshitz L., Rigid subanalytic sets, Amer. J. Math.115 (1993) 77-108. Zbl0792.14010MR1209235
  30. [30] Lipshitz L., Robinson Z., Rigid subanalytic subsets of the line and the plane, Amer. J. Math.118 (1996) 493-527. Zbl0935.14035MR1393258
  31. [31] Lipshitz L., Robinson Z., Rigid subanalytic subsets of curves and surfaces, J. London Math. Soc. (2)59 (1999) 895-921. Zbl0935.32008MR1709087
  32. [32] Lipshitz L., Robinson Z., Rings of separated power series, Astérisque264 (2000) 3-108. Zbl0957.32011MR1758887
  33. [33] Lipshitz L., Robinson Z., Model completeness and subanalytic sets, Astérisque264 (2000) 109-126. 
  34. [34] Lipshitz L., Robinson Z., Uniform properties of rigid subanalytic sets, Trans. Amer. Math. Soc.357 (1) (2005) 4349-4377. Zbl1081.03024MR2156714
  35. [35] Loeser F., Sebag J., Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J.119 (2) (2003) 315-344. Zbl1078.14029MR1997948
  36. [36] Macintyre A., Rationality of p-adic Poincaré series: uniformity in p, Ann. Pure Appl. Logic49 (1) (1990) 31-74. Zbl0731.12015MR1076249
  37. [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). Zbl1079.14005
  38. [38] Pas J., Uniform p-adic cell decomposition and local zeta-functions, Journal für die reine und angewandte Mathematik399 (1989) 137-172. Zbl0666.12014MR1004136
  39. [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. Zbl0659.12017MR1023804
  40. [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. Zbl1040.03031MR2018563
  41. [41] Sebag J., Rationalité des séries de Poincaré et des fonctions zêta motiviques, Manuscripta Math.115 (2) (2004) 125-162, (in French). Zbl1073.14524MR2098466
  42. [42] Sebag J., Intégration motivique sur les schémas formels, Bull. Soc. Math. France132 (1) (2004) 1-54, (in French). Zbl1084.14012MR2075915

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