Decomposition into special cubes and its applications to quasi-subanalytic geometry

Abstract

Abstract. The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.This paper deals with certain families of quasianalytic Q-functions as well as the corresponding categories Q of quasianalytic Q-manifolds and Qmappings. Transformation to normal crossings by blowing up applies to such Q-functions (as discovered by 3] and Rolin-Speissegger-Wilkie [13]), and thence to Q-semianalytic sets. This gives rise to the geometry of Q-subanalytic sets, which are a natural generalization of the classical subanalytic sets.Our main purpose is to present a decomposition of a relatively compact Q-semianalytic set into a finite union of special cubes, and of a relatively compact Q-subanalytic set into a finite number of immersion cubes. The former decomposition combines transformation to normal crossings by local blowing up (developed in [1,3]) and a suitable partitioning; together with the method of fiber cutting, it yields the latter decomposition. Decomposition into special cubes will also become a basic tool in our subsequent paper [11] concerning quantifier elimination and the preparation theorem in quasianalytic geometry.We apply decomposition into immersion cubes in our proof of Gabrielov's complement theorem for the case of Q-subanalytic sets. These two results both imply that the expansion R Q of the real field by restricted quasianalytic Q-functions is an o-minimal polynomially bounded structure with exponent 2000 Mathematics Subject Classification: Primary 14P15, 32B20, 26E10; Secondary 32S45, 03C64.

A theorem on generic intersections in an o-minimal structure

“…sets which are subanalytic or quasi-subanalytic in a semialgebraic compactification (see e.g. [11,7]). In turn, the locally definable sets are then precisely the subanalytic or quasisubanalytic ones.…”

Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices

“…Our proof of that theorem made use, however, of Luengo's statement that every quasiordinary Weierstrass polynomial in Tschirnhausen form is ν-quasiordinary in the sense of Hironaka. Therefore, those results of ours bear a relative character, because it turned out, as indicated in [9], that Luengo's proof seems to have an essential gap.As in our previous papers [14,15], we fix a family Q = (Q m ) m∈N of sheaves of local R-algebras of smooth functions on R m . For each open subset U ⊂ R m , Q(U ) = Q m (U ) is thus a subalgebra of the algebra C ∞ m (U ) of smooth real functions on U .…”

“…As in our previous papers [14,15], we fix a family Q = (Q m ) m∈N of sheaves of local R-algebras of smooth functions on R m . For each open subset U ⊂ R m , Q(U ) = Q m (U ) is thus a subalgebra of the algebra C ∞ m (U ) of smooth real functions on U .…”

“…Denote by R Q the expansion of R by restricted Q-functions, which is an ominimal polynomially bounded structure with exponent field Q and admits smooth quasianalytic cell decomposition (cf. [26,14]. From now on, the word "definable" means "definable (with parameters) in the structure R Q ".…”

On arc-analytic functions definable by a Weierstrass system

“…The structure R admits W-analytic cell decomposition, and thus finite Wanalytic stratifications of definable subsets too (see e.g. [4,5,8]). Therefore, one can partition the set U into finitely many definable W-analytic submanifolds M 1 , .…”