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A proof of the valuation property and preparation theorem

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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, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory $T_{conv}$ (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den Dries-Lewenberg. The main tools applied here are saturation, the Marker-Steinhorn theorem on parameter reduction and heir-coheir amalgams.
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. The latter originates in the work of Parusiński for subanalytic functions, and of Lion-Rolin for logarithmic-exponential functions. Van den Dries-Speissegger 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 ℝ with exponential function and logarithm.
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  • Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
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