Geometric categories and o-minimal structures.(English)Zbl 0889.03025

The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, E. Bierstone and P. D. Milman [Publ. Math., Inst. Hautes Étud. Sci. 67, 5-42 (1988; Zbl 0674.32002)]. Regrettably, certain “nice” sets – such as $$\{(x, x^r): x>0\}$$ for positive irrational $$r$$, and $$\{(x, e^{-1/x}): x>0\}$$ – are not subanalytic (at the origin) in $$\mathbb{R}^2$$. Here we make available an extension of the category of subanalytic sets that has these sets among its objects and that behaves much like the category of subanalytic sets. The possibility of doing this emerged in 1991 when A. Wilkie [J. Am. Math. Soc. 9, No. 4, 1051-1094 (1996)] proved that the real exponential field is “model complete”, followed soon by work of Ressayre, Macintyre, Marker and the authors. However, there are two obstructions to the use by geometers of this development: (i) while the proofs in these articles make essential use of model theory, many results are also stated there (efficiently, but unnecessarily) in model-theoretic terms; (ii) the results of these papers apply directly only to the Cartesian spaces $$\mathbb{R}^n$$, and not to arbitrary real analytic manifolds. Consequently, in order to carry out our goal, we recast here some results in those papers – as well as many of their consequences – in more familiar terms, with emphasis on results of a geometric nature, and allowing arbitrary (real analytic) manifolds as ambient spaces.

MSC:

 03C60 Model-theoretic algebra 32B20 Semi-analytic sets, subanalytic sets, and generalizations 14P99 Real algebraic and real-analytic geometry

Zbl 0674.32002
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References:

 [1] E. Bierstone and P. Milman, Semianalytic and subanalytic sets , Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5-42. · Zbl 0674.32002 [2] J. Bochnak, M. Coste, and M.-F. Roy, Géométrie algébrique réelle , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 12, Springer-Verlag, Berlin, 1987. · Zbl 0633.14016 [3] L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results , Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 189-193. · Zbl 0612.03008 [4] L. van den Dries, Tame Topology and $$O$$-Minimal Structures , monograph in preparation. [5] L. van den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation , Ann. of Math. (2) 140 (1994), no. 1, 183-205. · Zbl 0837.12006 [6] L. van den Dries, A. Macintyre, and D. Marker, Logarithmic-exponential power series , to appear in J. London Math. Soc. (2). · Zbl 0924.12007 [7] L. van den Dries and C. Miller, Extending Tamm’s theorem , Ann. Inst. Fourier (Grenoble) 44 (1994), no. 5, 1367-1395. · Zbl 0816.32004 [8] L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions , Israel J. Math. 85 (1994), no. 1-3, 19-56. · Zbl 0823.03017 [9] R. Hardt, Stratification of real analytic mappings and images , Invent. Math. 28 (1975), 193-208. · Zbl 0298.32003 [10] R. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings , Amer. J. Math. 102 (1980), no. 2, 291-302. JSTOR: · Zbl 0465.14012 [11] R. Hardt, Some analytic bounds for subanalytic sets , Differential geometric control theory (Houghton, Mich., 1982), Progr. Math., vol. 27, Birkhäuser Boston, Boston, MA, 1983, pp. 259-267. · Zbl 0547.32003 [12] M. Kashiwara and P. Schapira, Sheaves on manifolds , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. · Zbl 0709.18001 [13] T. Kayal and G. Raby, Ensembles sous-analytiques: quelques propriétés globales , C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 18, 521-523. · Zbl 0674.32003 [14] T. Loi, On the global Łojasiewicz inequalities for the class of analytic logarithmico-exponential functions , C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 6, 543-548. · Zbl 0804.32008 [15] T. Loi, Whitney stratification of sets definable in the structure $$\mathbbR_\exp$$ , to appear in Banach Center Publ. · Zbl 0904.14030 [16] S. Łojasiewicz, Ensembles semi-analytiques , Institut des Hautes Études Scientifiques, Bures-sur-Yvette, 1965. [17] J. Mather, Stratifications and mappings , Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) ed. M. M. Peixoto, Academic Press, New York, 1973, pp. 195-232. · Zbl 0286.58003 [18] C. Miller, Expansions of the real field with power functions , Ann. Pure Appl. Logic 68 (1994), no. 1, 79-94. · Zbl 0823.03018 [19] C. Miller, Exponentiation is hard to avoid , Proc. Amer. Math. Soc. 122 (1994), no. 1, 257-259. JSTOR: · Zbl 0808.03022 [20] C. Miller, Infinite differentiability in polynomially bounded o-minimal structures , Proc. Amer. Math. Soc. 123 (1995), no. 8, 2551-2555. JSTOR: · Zbl 0823.03019 [21] J.-P. Ressayre, Integer parts of real closed exponential fields (extended abstract) , Arithmetic, proof theory, and computational complexity (Prague, 1991), Oxford Logic Guides, vol. 23, Oxford Univ. Press, New York, 1993, pp. 278-288. · Zbl 0791.03018 [22] W. Schmid and K. Vilonen, Characteristic cycles of constructible sheaves , Invent. Math. 124 (1996), no. 1-3, 451-502. · Zbl 0851.32011 [23] M. Shiota, Nash manifolds , Lecture Notes in Mathematics, vol. 1269, Springer-Verlag, Berlin, 1987. · Zbl 0629.58002 [24] M. Shiota, Geometry of subanalytic and semialgebraic sets: abstract , Real analytic and algebraic geometry (Trento, 1992), de Gruyter, Berlin, 1995, pp. 251-275. · Zbl 0870.32001 [25] M. Tamm, Subanalytic sets in the calculus of variation , Acta Math. 146 (1981), no. 3-4, 167-199. · Zbl 0478.58010 [26] J.-Cl. Tougeron, Algèbres analytiques topologiquement noethériennes. Théorie de Khovanskii , Ann. Inst. Fourier (Grenoble) 41 (1991), no. 4, 823-840. · Zbl 0786.32011 [27] A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function , to appear in J. Amer. Math. Soc. JSTOR: · Zbl 0892.03013
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