We fix an o-minimal expansion of the real field, M say. Definabilitynotions are with respect to M. Let F = {f_x : x in X} be a definable familyof (single valued) complex analytic functions, each one having domain somedisk, D_x say, in ?, where the parameter space X is a definable subset of ?^mfor some m. We present some finiteness theorems for such families F whichare uniform in parameters and give some applications.
We also speculate on the notion of “definable” Riemann surface.
- Séminaire Géométrie et théorie des modèles