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Uniform properties of rigid subanalytic sets

Author(s): Leonard Lipshitz; Zachary Robinson
Journal: Trans. Amer. Math. Soc. 357 (2005), 4349-4377.
MSC (2000): Primary 03C10, 32P05, 32B20; Secondary 26E30, 03C98
Posted: June 21, 2005
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Abstract: In the context of rigid analytic spaces over a non-Archimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the field; in particular, the analytic quantifier elimination is independent of the valued field's characteristic, residue field and value group, in close analogy to the algebraic case. This provides uniformity results about rigid subanalytic sets. We obtain uniform versions of smooth stratification for subanalytic sets and the \Lojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set.

References:

[Ba1]
W. Bartenwerfer, Der allgemeine Kontinuitätssatz für $k$-meromorphe Funktionen im Di-zylinder, Math. Ann. 191 (1971), 196-234. MR 0287035 (44:4242)

[Ba2]
W. Bartenwerfer, Die Beschränktheit der Stückzahl der Fasern $K$-analytischer Abbildungen, J. Reine Angew. Math. 416 (1991), 49-70. MR 1099945 (92e:32017)

[BGR]
S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean Analysis, Springer-Verlag, 1984. MR 0746961 (86b:32031)

[D1]
J. Denef, $p$-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math. 369 (1987), 991-1008. MR 0850632 (88d:11030)

[D2]
-, Arithmetic and geometric applications of quantifier elimination for valued fields, Model Theory, Algebra and Geometry, vol. 39, MSRI Publications, 2000, pp. 173-198. MR 1773707 (2001e:03063)

[DD]
J. Denef and L. van den Dries, $p$-adic and real subanalytic sets, Ann. Math. 128 (1988), 79-138. MR 0951508 (89k:03034)

[E]
O. Endler, Valuation Theory, Springer-Verlag, 1972. MR 0357379 (50:9847)

[H]
R. Huber, On semianalytic subsets of rigid analytic varieties, Geometriae Dedicata 58 (1995), 291-311. MR 1358458 (96k:32069)

[Kap1]
I. Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 312-321. MR 0006161 (3:264d)

[Kap2]
-, Maximal fields with valuations II, Duke Math. J. 12 (1945), 243-248. MR 0012276 (7:3d)

[L]
L. Lipshitz, Rigid subanalytic sets, Amer. J. Math. 115 (1993), 77-108. MR 1209235 (94d:32047)

[Liu]
N. Liu, Analytic cell decomposition and the closure of $p$-adic semianalytic sets, JSL 62 (1997), 285-303. MR 1450524 (98h:03039)

[LR1]
L. Lipshitz and Z. Robinson, Rigid subanalytic subsets of the line and the plane, Amer. J. Math. 118 (1996), 493-527. MR 1393258 (97e:14067)

[LR2]
-, Rigid subanalytic subsets of curves and surfaces, J. London Math. Soc. (2) 59 (1999), 895-921. MR 1709087 (2000j:32037)

[LR3]
-, One-dimensional fibers of rigid subanalytic sets, J. Symbolic Logic 63 (1998), 83-88. MR 1610778 (98m:32049)

[LR4]
-, Rings of separated power series, Astérisque 264 (2000), 3-108. MR 1758887 (2001g:32017)

[LR5]
-, Model completeness and subanalytic sets, Astérisque 264 (2000), 109-126.

[LR6]
-, Quasi-affinoid varieties, Astérisque 264 (2000), 127-149.

[LR7]
-, Dimension theory and smooth stratification of rigid subanalytic sets, Logic Conference '98 (S. Buss, P. Hájek and P. Pudlak, eds.); Lecture Notes in Logic, vol. 13, A.S.L., 2000, pp. 302-315. MR 1743267 (2002b:32014)

[Mac1]
A. Macintyre, On definable subsets of $p$-adic fields, J. Symbolic Logic 41 (1976), 605-610. MR 0485335 (58:5182)

[Mac2]
-, Rationality of $p$-adic Poincaré series: uniformity in $p$, Ann. Pure Appl. Logic 49 (1990), 31-74. MR 1076249 (92b:11085)

[Mat]
H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989. MR 1011461 (90i:13001)

[Pas]
J. Pas, Uniform $p$-adic cell decomposition and local zeta functions, J. Reine Angew. Math. 399 (1989), 137-172. MR 1004136 (91g:11142)

[Pass]
D.S. Passman, The Algebraic Structure of Group Rings, J. Wiley, 1977. MR 0470211 (81d:16001)

[Po]
B. Poonen, Maximally Complete Fields, L'Enseignement Math. 39 (1993), 87-106. MR 1225257 (94h:12005)

[Ro]
A. Robinson, Complete Theories, North-Holland, 1956. MR 0075897 (17:817b)

[Rob1]
Z. Robinson, Smooth points of $p$-adic subanalytic sets, Manuscripta Math. 80 (1993), 45-71. MR 1226596 (94h:32055)

[Rob2]
-, Flatness and smooth points of $p$-adic subanalytic sets, Ann. Pure Appl. Logic 88 (1997), 217-225. MR 1600911 (99b:32050)

[S1]
H. Schoutens, Rigid subanalytic sets, Comp. Math. 94 (1994), 269-295. MR 1310860 (96b:32043)

[S2]
-, Rigid subanalytic sets in the plane, J. Algebra 170 (1994), 266-276. MR 1302840 (96b:32041)

[S3]
-, Uniformization of rigid subanalytic sets, Comp. Math. 94 (1994), 227-245. MR 1310858 (96b:32042)

[S4]
-, Blowing up in rigid analytic geometry, Bull. Belg. Math. Soc. 2 (1995), 399-417. MR 1355829 (96g:32053)

[S5]
-, Closure of rigid semianalytic sets, J. Algebra 198 (1997), 120-134. MR 1482978 (98m:14054)

[vdD]
L. van den Dries, Analytic Ax-Kochen-Ersov Theorems, Contemporary Mathematics, vol. 131 part 3, AMS, 1992, pp. 379-398. MR 1175894 (93i:03041)

[vHM]
L. van den Dries, D. Haskell and D. Macpherson, One-dimensional $p$-adic subanalytic sets, J. London Math. Soc. (2) 59 (1999), 1-20. MR 1688485 (2000k:03077)

[W]
V. Weispfenning, Quantifier Elimination and Decision Procedures for Valued Fields, Models and Sets, Aachen 1103 (1983), 1984, pp. 419-472. MR 0775704 (86m:03059)

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Additional Information:

Leonard Lipshitz
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: lipshitz@math.purdue.edu

Zachary Robinson
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: robinsonz@mail.ecu.edu

DOI: 10.1090/S0002-9947-05-04003-1
PII: S 0002-9947(05)04003-1
Received by editor(s): March 7, 2003
Posted: June 21, 2005
Additional Notes: This work was supported in part by NSF grant number DMS 0070724
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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