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 (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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap92-1-8,
author = {Krzysztof Jan Nowak},
title = {A proof of the valuation property and preparation theorem},
journal = {Annales Polonici Mathematici},
volume = {92},
year = {2007},
pages = {75-85},
zbl = {1143.03019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap92-1-8}
}
Krzysztof Jan Nowak. A proof of the valuation property and preparation theorem. Annales Polonici Mathematici, Tome 92 (2007) pp. 75-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap92-1-8/