Uniform properties of rigid subanalytic sets

Lipshitz, Leonard; Robinson, Zachary

doi:10.1090/S0002-9947-05-04003-1

Uniform properties of rigid subanalytic sets
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by Leonard Lipshitz and Zachary Robinson

Trans. Amer. Math. Soc. 357 (2005), 4349-4377

DOI: https://doi.org/10.1090/S0002-9947-05-04003-1

Published electronically: 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 Łojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set.

References

Wolfgang Bartenwerfer, Der allgemeine Kontinuitätssatz für $k$-meromorphe Funktionen im Dizylinder, Math. Ann. 191 (1971), 196–234 (German). MR 287035, DOI 10.1007/BF01578708
Wolfgang Bartenwerfer, Die Beschränktheit der Stückzahl der Fasern $K$-analytischer Abbildungen, J. Reine Angew. Math. 416 (1991), 49–70 (German). MR 1099945, DOI 10.1515/crll.1991.416.49
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
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
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
Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379, DOI 10.1007/978-3-642-65505-0
Roland Huber, On semianalytic subsets of rigid analytic varieties, Geom. Dedicata 58 (1995), no. 3, 291–311. MR 1358458, DOI 10.1007/BF01263458
Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
Irving Kaplansky, Maximal fields with valuations. II, Duke Math. J. 12 (1945), 243–248. MR 12276
L. Lipshitz, Rigid subanalytic sets, Amer. J. Math. 115 (1993), no. 1, 77–108. MR 1209235, DOI 10.2307/2374723
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
Leonard Lipshitz and Zachary Robinson, Rigid subanalytic subsets of the line and the plane, Amer. J. Math. 118 (1996), no. 3, 493–527. MR 1393258, DOI 10.1353/ajm.1996.0027
Leonard Lipshitz and Zachary Robinson, Rigid subanalytic subsets of curves and surfaces, J. London Math. Soc. (2) 59 (1999), no. 3, 895–921. MR 1709087, DOI 10.1112/S0024610799007358
L. Lipshitz and Z. Robinson, One-dimensional fibers of rigid subanalytic sets, J. Symbolic Logic 63 (1998), no. 1, 83–88. MR 1610778, DOI 10.2307/2586589
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
—, Model completeness and subanalytic sets, Astérisque 264 (2000), 109–126.
—, Quasi–affinoid varieties, Astérisque 264 (2000), 127–149.
Leonard Lipshitz and Zachary Robinson, Dimension theory and smooth stratification of rigid subanalytic sets, Logic Colloquium ’98 (Prague), Lect. Notes Log., vol. 13, Assoc. Symbol. Logic, Urbana, IL, 2000, pp. 302–315. MR 1743267
Angus Macintyre, On definable subsets of $p$-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610. MR 485335, DOI 10.2307/2272038
Angus Macintyre, Rationality of $p$-adic Poincaré series: uniformity in $p$, Ann. Pure Appl. Logic 49 (1990), no. 1, 31–74. MR 1076249, DOI 10.1016/0168-0072(90)90050-C
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
Johan Pas, Uniform $p$-adic cell decomposition and local zeta functions, J. Reine Angew. Math. 399 (1989), 137–172. MR 1004136, DOI 10.1515/crll.1989.399.137
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
Bjorn Poonen, Maximally complete fields, Enseign. Math. (2) 39 (1993), no. 1-2, 87–106. MR 1225257
Abraham Robinson, Complete theories, North-Holland Publishing Co., Amsterdam, 1956. MR 0075897
Zachary Robinson, Smooth points of $p$-adic subanalytic sets, Manuscripta Math. 80 (1993), no. 1, 45–71. MR 1226596, DOI 10.1007/BF03026536
Zachary Robinson, Flatness and smooth points of $p$-adic subanalytic sets, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 217–225. Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR 1600911, DOI 10.1016/S0168-0072(97)00023-7
Hans Schoutens, Rigid subanalytic sets, Compositio Math. 94 (1994), no. 3, 269–295. MR 1310860
Hans Schoutens, Rigid subanalytic sets in the plane, J. Algebra 170 (1994), no. 1, 266–276. MR 1302840, DOI 10.1006/jabr.1994.1337
Hans Schoutens, Uniformization of rigid subanalytic sets, Compositio Math. 94 (1994), no. 3, 227–245. MR 1310858
Hans Schoutens, Blowing up in rigid analytic geometry, Bull. Belg. Math. Soc. Simon Stevin 2 (1995), no. 4, 399–417. MR 1355829
Hans Schoutens, Closure of rigid semianalytic sets, J. Algebra 198 (1997), no. 1, 120–134. MR 1482978, DOI 10.1006/jabr.1997.7142
Lou van den Dries, Analytic Ax-Kochen-Ersov theorems, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 379–398. MR 1175894
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
Volker Weispfenning, Quantifier elimination and decision procedures for valued fields, Models and sets (Aachen, 1983) Lecture Notes in Math., vol. 1103, Springer, Berlin, 1984, pp. 419–472. MR 775704, DOI 10.1007/BFb0099397