Real Algebraic and Analytic Geometry
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169. Krzysztof Jan Nowak:
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.

Full text, 14p.: dvi 48k, ps.gz 141k, pdf 179k.