[Algebraic relations between values of Siegel -functions and Mahler -functions]
We show that all algebraic relations over between the values of Siegel -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 between the functions under consideration. We obtain a similar result for the Mahler -functions, in which the algebro-differential relations are replaced by the -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 -functions and that of -functions.
Nous montrons que toutes les relations algébriques sur entre les valeurs prises par des -fonctions de Siegel en un point algébrique non nul sont d’origine fonctionnelle, en ce sens qu’elles s’obtiennent par dégénérescence de relations algébro-différentielles sur entre les fonctions considérées. Nous obtenons un résultat analogue pour les -fonctions de Mahler, dans lequel les relations dites -algébriques se substituent aux relations algébro-différentielles. Nous donnons également plusieurs conséquences de ce résultat, notamment concernant certains phénomènes de descente. Le point de vue adopté révèle des similitudes frappantes entre la théorie des -fonctions et celle des -fonctions.
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Mots-clés : Transcendance, indépendance algébrique, $E$-fonctions de Siegel, $M$-fonctions de Mahler
Keywords: Transcendence, algebraic independence, Siegel $E$-functions, Mahler $M$-functions
Boris Adamczewski 1; Colin Faverjon 1
@article{CRMATH_2024__362_G10_1215_0, author = {Boris Adamczewski and Colin Faverjon}, title = {Relations alg\'ebriques entre valeurs de $E$-fonctions ou de $M$-fonctions}, journal = {Comptes Rendus. Math\'ematique}, pages = {1215--1241}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.634}, language = {fr}, }
TY - JOUR AU - Boris Adamczewski AU - Colin Faverjon TI - Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions JO - Comptes Rendus. Mathématique PY - 2024 SP - 1215 EP - 1241 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.634 LA - fr ID - CRMATH_2024__362_G10_1215_0 ER -
Boris Adamczewski; Colin Faverjon. Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1215-1241. doi : 10.5802/crmath.634. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.634/
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