Abstract :
We describe here the state of the art in transcendence methods, proving in an abstract setting two general theorems which can be used as an “algorithm” to establish transcendence of values of analytic functions. Combining these theorems with effective zero estimates gives measures of approximation and, using further approximation properties, lead to algebraic independence results (presently up to transcendence degree 3 or even 4, depending on the situation). In a second part we give several examples of application of the method to known, new, and (hopefully) future results. This includes transcendence properties of the invariant modular function and of Mahler type functions, algebraic independence of values of Eisenstein series, algebraic independence of values of exponentials of Drinfeld modules, and a strategy to tackle the Gelfond–Schneider conjecture.
