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o-minimal preparation theorems. (English) Zbl 1081.03039

BĂ©lair, L. (ed.) et al., Model theory and applications. Based on the Euro-Conference in model theory and applications, Ravello, Italy, May 27–June 1, 2002. Rome: Aracne (ISBN 88-7999-411-5/hbk). Quaderni di Matematica 11, 87-116 (2002).
From the introduction: The first author, A. Macintyre, and D. Marker [Ann. Math. (2) 140, 183–205 (1994; Zbl 0837.12006)] used model theory to show that globally subanalytic functions are piecewise given by terms in a certain language. Adding symbols for the exponential and logarithm function to this language, they also showed that functions definable from exponentiation and globally subanalytic functions are piecewise given by terms. This piecewise definability by terms inspired J.-M. Lion and J.-P. Rolin [Ann. Inst. Fourier 47, 859–884 (1997; Zbl 0873.32004)] to find geometric proofs for what they call preparation theorems, which are sometimes more useful in applications.
In Section 2 we prove a preparation theorem for functions belonging to a polynomially bounded o-minimal structure. In the later sections we extend this to functions that are logarithmic-exponential over a polynomially bounded o-minimal structure.
For the entire collection see [Zbl 1066.03014].

MSC:

03C64 Model theory of ordered structures; o-minimality
03C10 Quantifier elimination, model completeness, and related topics
32B05 Analytic algebras and generalizations, preparation theorems
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