The Siegel-Shidlovskii Method Revisited

We revisit the classical Siegel-Shidlovskii method in transcendental number theory from a new perspective, in which functional relations play a central role. Within a single self-contained framework, we recover the foundational results in the theory of E-functions, including the Siegel-Shidlovskii theorem, Beukers' lifting theorem, and Andr{é}'s theorem on the singularities of the minimal differential operator annihilating an E-function. Notably, the latter is obtained without any recourse to the theory of Gfunctions. This unified approach also allows us to resolve a problem posed by Shidlovskii: we obtain a general algebraic independence measure extending the classical Siegel-Lang measure to an arbitrary algebraically independent family of E-values. We also provide a short elementary proof of a second theorem of Andr{é}, which has become an important tool in the theory of Efunctions and concerns the primality of the specialization, at a regular point, of the ideal of functional relations of a vector of E-functions satisfying a differential system.