Real Algebraic and Analytic Geometry |

Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings.

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Submission: 2008, May 19.

*Abstract:
This paper investigates the geometry of the
expansion ${\cal R}_{Q}$ of the real field $\matR$ by restricted
quasianalytic functions. The main purpose is to establish
quantifier elimination, description of definable functions by
terms, the valuation property and preparation theorem (in the
sense of Parusi\'nski--Lion--Rolin). To this end, we study
non-standard models $\cal R$ of the universal diagram $T$ of
${\cal R}_{Q}$ in the language $\cal L$ augmented by the names of
rational powers. Our approach makes no appeal to the Weierstrass
preparation theorem, upon which majority of fundamental results in
analytic geometry rely, but which is unavailable in the general
quasianalytic geometry. The basic tools applied here are
transformation to normal crossings and decomposition into special
cubes. The latter method, developed in our article~\cite{Now1},
combines modifications by blowing up with a suitable partitioning.
Via an analysis of $\cal L$-terms and infinitesimals, we prove the
valuation property for functions given by $\cal L$-terms, and next
the exchange property for substructures of a given model $\cal R$.
Our proofs are based on the concepts of analytically independent
as well as active and non-active infinitesimals, introduced in
this article. Further, quantifier elimination for $T$ is
established through model-theoretic compactness. The universal
theory $T$ is thus complete and o-minimal, and ${\cal R}_{Q}$ is
its prime model. Under the circumstances, every definable function
is given piecewise by $\cal L$-terms, and therefore the previous
results concerning $\cal L$-terms generalize immediately to
definable functions. In this fashion, we obtain the valuation
property and preparation theorem for quasi-subanalytic functions.
Finally, a quasi-subanalytic version of Puiseux's theorem with
parameter is demonstrated.*

Mathematics Subject Classification (2000): 32S45, 14P15, 32B20, 03C10, 26E10, 03C64.

Keywords and Phrases: quasianalytic functions, special cubes, special modifications, analytically independent infinitesimals, active and non-active infinitesimals, valuation property, quantifier elimination, preparation theorem.

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