Given two (real or complex) analytic functions f and g, we aim to understand under which hypotheses f and g are locally interdefinable, in the context of o-minimal structures. Local interdefinability generalises (differential) algebraic dependence: locally interdefinable functions satisfy relations expressible by some first-order formulas, which are in general more complicated than differential algebraic relations.

There are situations where differential algebraic independence implies independence with respect to local definability: it is the case for complex exponentiation and any Weierstrass p-function.

There are however holomorphic functions which are locally interdefinable without being differentially algebraically dependent, and which cannot be obtained from one another by elementary operations such as composition and extracting implicit functions.

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. This is a special case of a more general result on quasianalytic classes.

I will discuss the aforementioned results and their proofs, which rely on an interaction between methods from functional transcendence, resolution of singularities and model theory.