Appears in collection : Model Theory and Applications

Given two (real or complex) analytic functions f and g, it is not sensible in general to ask whether they are first-order interdefinable as total functions (think of the sine function). It does make sense to ask whether f and g are locally interdefinable in the context of o-minimal structures. For example, the real exponential function and the sine function are not locally interdefinable [Bianconi]. The same holds for complex exponentiation and any Weierstrass ℘-function [Jones, Kirby, Servi]. Two Weierstrass ℘-functions are locally interdefinable if and only if one can be obtained from the other by isogeny and Schwarz reflection [Jones, Kirby, Servi]. There are complex analytic functions which are locally interdefinable and which cannot be obtained from one another by elementary operations such as Schwarz reflection, composition, derivation and extracting implicit functions [Jones, Kirby, Le Gal, Servi]. In the case of real analytic functions, it is possible to give an analytic characterisation of all the functions g which are locally definable from f [Le Gal, Servi, Vieillard-Baron]. The proofs of the aforementioned results rely on the interaction between methods from functional transcendence, resolution of singularities and model theory.