Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions - Adamczewski, Boris; Faverjon, Colin (Comptes Rendus. Mathématique)
We show that all algebraic relations over $\overline{\mathbb{Q}}$ between the values of Siegel $E$-functions at non-zero algebraic points have a functional origin, in the sense that they can be obtained by degeneracy of algebro-differential relations over $\overline{\mathbb{Q}}(z)$ between the functions under consideration. We obtain a similar result for the Mahler $M_q$-functions, in which the algebro-differential relations are replaced by the $σ _q$-algebraic relations. We also give several consequences of this result, in particular with respect to certain descent phenomena. The point of view adopted reveals striking similarities between the theory of $E$-functions and that of $M_q$-functions.