A proof of the valuation property and preparation theorem

• Autorzy: Krzysztof Jan Nowak
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 32B20: Semi-analytic sets and subanalytic sets
• 14P15: Real analytic and semianalytic sets
• 03C64: Model theory of ordered structures; o-minimality

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2007
• Tom: 92
• Numer: 1
• Strony: 75-85

Daty

wydano

2007

Twórcy

autor

Krzysztof Jan Nowak

• Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Identyfikatory

DOI

10.4064/ap92-1-8