A proof of the valuation property and preparation theorem

Krzysztof Jan Nowak. "A proof of the valuation property and preparation theorem." Annales Polonici Mathematici 92.1 (2007): 75-85. <http://eudml.org/doc/281119>.

@article{KrzysztofJanNowak2007,
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 $T_\{conv\}$ (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.},
author = {Krzysztof Jan Nowak},
journal = {Annales Polonici Mathematici},
keywords = {o-minimal structures; valuation property; preparation theorem},
language = {eng},
number = {1},
pages = {75-85},
title = {A proof of the valuation property and preparation theorem},
url = {http://eudml.org/doc/281119},
volume = {92},
year = {2007},
}

TY - JOUR
AU - Krzysztof Jan Nowak
TI - A proof of the valuation property and preparation theorem
JO - Annales Polonici Mathematici
PY - 2007
VL - 92
IS - 1
SP - 75
EP - 85
AB - 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.
LA - eng
KW - o-minimal structures; valuation property; preparation theorem
UR - http://eudml.org/doc/281119
ER -