Real Algebraic and Analytic Geometry |
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Submission: 2012, June 26.
Abstract:
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.
Full text, 24p.: dvi 81k, pdf 313k.