Real Algebraic and Analytic Geometry |

A proof of the valuation property and preparation theorem.

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Submission: 2006, March 17.

*Abstract:
The purpose of this article is to present a short model-theoretic
proof of the valuation property for a polynomially bounded,
o-minimal theory $T$. The valuation property was conjectured by
van den Dries~\cite{1}, and proved for the polynomially bounded
case by van den Dries--Speissegger~\cite{4} and for the power
bounded case by Tyne~\cite{11}.
Our proof uses the transfer principle for the theory $T_{conv}$
(theory $T$ with an extra unary symbol denoting a proper convex
subring) which
--- together with quantifier elimination --- is due to van den
Dries--Lewenberg~\cite{2}. The main tools applied here are
saturation, the Marker--Steinhorn theorem on parameter
reduction~\cite{8} and heir-coheir amalgams (see e.g.~\cite{6},
Chap.~6).
The significance of the valuation property lies to a great extent
in its geometric content: it is equivalent to the preparation
theorem (which says, roughly speaking, that every definable
function of several variables depends piecewise on any fixed
variable in a certain simple fashion). This theorem originates in
Parusi\'{n}ski~\cite{9,10} for subanalytic functions, and in
Lion--Rolin~\cite{7} for logarithmic-exponential functions. Van
den Dries--Speissegger~\cite{5} have proved the preparation
theorem in the o-minimal setting (for functions definable in a
polynomially bounded structure or logarithmic-exponential over
such a structure). Also, the valuation property makes it possible
to establish quantifier elimination for polynomially bounded
expansions of the real field $\matR$ with exponential function and
logarithm (see \cite{4,3}).*

Mathematics Subject Classification (2000): 03C64, 12J25, 14P15.

Keywords and Phrases: o-minimal structures, valuation property, preparation theorem.

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