Geometric categories and o-minimal structures
The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, \textit{E. Bierstone} and \textit{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 \textit{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.
extension of the category of subanalytic sets
scientific article; zbMATH DE number 926955
scientific article