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.

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton–Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

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Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (ℛ, V) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.