A proof of the valuation property and preparation theorem

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 [1], and proved for the polynomially bounded case by van den Dries– Speissegger [4] and for the power bounded case by Tyne [11]. Our proof uses the transfer principle for the theory Tconv (theory T with an extra unary symbol denoting a proper convex subring) which — together with quantifier elimination — is due to van den Dries–Lewenberg [2]. The main tools applied here are saturation, the Marker–Steinhorn theorem on parameter reduction [8] and heir-coheir amalgams (see e.g. [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 Parusiński [9, 10] for subanalytic functions, and in Lion–Rolin [7] for logarithmic-exponential functions. Van den Dries–Speissegger [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 R with exponential function and logarithm (see [4, 3]). 2001 Mathematics Subject Classification: 03C64, 12J25, 14P15.