Real Algebraic and Analytic Geometry
Submission: 2012, June 26.
This paper continues our previous article devoted to quantifier elimination and the valuation property for the expansion of the real field by restricted quasianalytic functions. A basic tool developed there was the concept of active and non-active infinitesimals, whose study relied on transformation to normal crossings by blowing up, and the technique of special cubes and modifications, introduced in our earlier papers. However, the theorem on an active infinitesimal, being one of the crucial results, was proved not in full generality (covering, nevertheless, the classical case of analytic functions). The main purpose of this paper is to provide a proof of the general quasianalytic case and, consequently, to legitimize the results of our previous article. Also given is yet another approach to quantifier elimination and a description of definable functions by terms (in the language augmented by the names of rational powers), which is much shorter and more natural with regard to the techniques applied. Finally, we present some theorems on the rectilinearization of definable functions, which are counterparts of those from our paper about functions definable by a Weierstrass system.
Mathematics Subject Classification (2010): 03C10, 14P15, 32S45; Secondary 03C64, 26E10, 32B20.
Keywords and Phrases: quasianalytic structures, quantifier elimination, active infinitesimals, special cubes and modifications, valuation property, rectilinearization of quasi-subanalytic functions.
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