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arXiv · 2303.05997

Relations alg\'ebriques entre valeurs de E-fonctions ou de M-fonctions

Boris Adamczewski·Colin Faverjon
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Abstract

We prove that all algebraic relations over $\overline{\mathbb Q}$ between values of Siegel's $E$-functions at some non-zero algebraic point have a functional source, in that they can be obtained as degeneration of $\delta$-algebraic relations over $\overline{\mathbb Q}(z)$ between the functions at stake. We also obtain an analogous result when considering Mahler's $M_q$-functions. In this case, the so-called $\sigma_q$-algebraic relations substitute for the $\delta$-algebraic relations. We also give several consequences of this result concerning some descent phenomena. The point of view we adopt here reveals some striking similarities between the theory of $E$-functions and the one of $M_q$-functions.

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Boris Adamczewski, Colin Faverjon. 2023-03-10. Relations alg\'ebriques entre valeurs de E-fonctions ou de M-fonctions. https://arxiv.org/abs/2303.05997

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