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

This paper deals with certain families of quasianalytic Q-functions. We present a decomposition of a relatively compact Q-semianalytic and a Q-subanalytic set into a finite union of special cubes and immersion cubes, respectively. Next, we prove Gabrielov’s complement theorem for the case of Q-subanalytic sets. Also derived are other fundamental properties of the expansion of the real field R by restricted quasianalytic Q-functions. This paper deals with certain families of quasianalytic Q-functions as well as the corresponding categories Q of quasianalytic Q-manifolds and Q-mappings. Transformation to normal crossings by blowing up applies to such Q-functions (as discovered by Bierstone–Milman [2, 3] and Rolin– Speissegger–Wilkie [18]), and thence to Q-semianalytic sets. This gives rise to the geometry of Q-subanalytic sets being 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 (based on transformation to normal crossings by local blowing up [1, 3] and a suitable partitioning) along with the method of fiber cutting yields the latter. Decomposition into special cubes will also become a basic tool in our subsequent paper [16]. Research partially supported by KBN Grant 1P03A00527. AMS Classification: Primary 14P15, 32B20, 26E10; Secondary 32S45, 03C64.