A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak¹

¹ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Krak/ow, Poland

Annales Polonici Mathematici, Tome 92 (2007) no. 1, pp. 75-85

Résumé

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.

Détail

DOI : 10.4064/ap92-1-8

Keywords: purpose article present short model theoretic proof valuation property polynomially bounded o minimal theory valuation property conjectured van den dries proved polynomially bounded van den dries speissegger power bounded tyne proof uses transfer principle theory conv extra unary symbol denoting proper convex subring which together quantifier elimination due van den dries lewenberg main tools applied here saturation marker steinhorn theorem parameter reduction heir coheir amalgams significance valuation property lies great extent its geometric content equivalent preparation theorem which says roughly speaking every definable function several variables depends piecewise fixed variable certain simple fashion latter originates work parusi nski subanalytic functions lion rolin logarithmic exponential functions van den dries speissegger have proved preparation theorem o minimal setting functions definable polynomially bounded structure logarithmic exponential structure valuation property makes possible establish quantifier elimination polynomially bounded expansions real field mathbb exponential function logarithm