Real Algebraic and Analytic Geometry
Submission: 2003, September 4.
We prove an effective version of Wilkie's theorem of the complement: in particular given an expansion of the real ordered field with finitely many C ∞-functions, if there are uniform and computable upper bounds on the number of connected components of quantifier free definable sets, then there are uniform and computable bounds for all definable sets. In such a case the theory of the structure is effectively o-minimal. We apply our results to the open problem of the decidability of the theory of the real field with the exponential function.
Mathematics Subject Classification (2000): 03C64, 03B25, 58A17.
Keywords and Phrases: o-minimality, Pfaffian functions, real exponentiation.
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