A proof of the valuation property and preparation theorem
Volume 92 / 2007
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, 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 $T_{\rm conv}$ (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 Parusi/nski 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 $\mathbb R$ with exponential function and logarithm.
