Geometric categories and o-minimal structures

Abstract

LOU VAN DEN DRIES AriD CHRIS MILLER Introduction. The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain "nice" sets-such as {(x, xr):x > 0} for positive irrational r, and { (x, e-1/x): x > 0}mare not subanalytic (at the origin) in IR2. 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 Wilkie [27] proved that the real exponential field is "model complete," followed soon by work of Ressayre, Macintyre, Marker and the authors; see [21], [5], [8], and [19]. 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 IRn, and not to arbitrary real analytic manifolds. Consequently, in order to carry out our goal, we recast here some results in those papersmas well as many of their consequencesmin more familiar terms, with emphasis on results of a geometric nature, and allowing arbitrary (real analytic) manifolds as ambient spaces. We thank W. Schmid and K. Vilonen for their suggestion that this would be a useful undertaking. Indeed, they gave us a "wish list" (inspired by Chapters 8 and 9 of Kashiwara and Schapira [12]; see also 10 of [22]) that strongly influenced the form and content of this paper. We axiomatize in Section 1 the notion of "behaving like the category of subanalytic sets" by introducing the notion of "analytic-geometric category." (The category Can of subanalytic sets is the "smallest" analytic-geometric category.) We also state in Section 1 a number of properties shared by all analyticgeometric categories. Proofs of the more difficult results of this nature, like the Whitney-stratifiability of sets and maps in such a category, often involve the use of charts to reduce to the case of subsets of IRn. For subsets of IRn, there already exists the theory of "o-minimal structures on the real field" (defined in Section 2). This subject is developed in detail in [4] and is an abstraction of the theory of semialgebraic sets (see, e.g., Bochnak et al. [2]). Each analytic-geometric category arises in a natural way from an o-minimal structure on the real field.

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

“…To obtain convergence, the only nontrivial fact that has to be checked is that f = λ x p + 1 2 Ax − b 2 is a KL function. For this, we recall that there exists a (polynomially bounded) o-minimal structure that contains the family of functions {x α : x > 0, α ∈ R} and restricted analytic functions (see [31,Example (5), p. 505 and Property 5.2 p. 513]). As a consequence, the results of [17] apply and f is a KL function with a desingularizing function of the form ϕ(s) = cs θ where c > 0, θ ∈ [0, 1).…”

Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity

“…Note that the Lojasiewicz inequality corresponds to the case ϕ(t) = t 1−ρ . In finite-dimensional spaces it has been shown in [37] that (1) is satisfied by a much larger class of functions, namely, by those that are definable in an o-minimal structure [17], or even more generally by functions belonging to analytic-geometric categories [25]. In the meantime the original Lojasiewicz result was used to derive new results in the asymptotic analysis of nonlinear heat equations [51], [35] and damped wave equations [30].…”

The normal cycle of a compact definable set

“…Our construction only uses easy results about subanalytic sets, like cell-decompositions and finiteness properties. In fact, the construction also works for all sets which belong to some o-minimal structure in the sense of [28] (they will be called definable for short) and even for so-called constructible functions.…”