Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions
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 funct...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-11-01
|
| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.634/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206175014060032 |
|---|---|
| author | Adamczewski, Boris Faverjon, Colin |
| author_facet | Adamczewski, Boris Faverjon, Colin |
| author_sort | Adamczewski, Boris |
| collection | DOAJ |
| description | 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 $\sigma _q$-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. |
| format | Article |
| id | doaj-art-a6d96b9be9cb4ab484c5f2f56d67e35f |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-a6d96b9be9cb4ab484c5f2f56d67e35f2025-02-07T11:23:31ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G101215124110.5802/crmath.63410.5802/crmath.634Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctionsAdamczewski, Boris0Faverjon, Colin1Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne Cedex, FranceUniv Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne Cedex, FranceWe 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 $\sigma _q$-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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.634/Transcendanceindépendance algébrique$E$-fonctions de Siegel$M$-fonctions de Mahler |
| spellingShingle | Adamczewski, Boris Faverjon, Colin Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions Comptes Rendus. Mathématique Transcendance indépendance algébrique $E$-fonctions de Siegel $M$-fonctions de Mahler |
| title | Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions |
| title_full | Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions |
| title_fullStr | Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions |
| title_full_unstemmed | Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions |
| title_short | Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions |
| title_sort | relations algebriques entre valeurs de e fonctions ou de m fonctions |
| topic | Transcendance indépendance algébrique $E$-fonctions de Siegel $M$-fonctions de Mahler |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.634/ |
| work_keys_str_mv | AT adamczewskiboris relationsalgebriquesentrevaleursdeefonctionsoudemfonctions AT faverjoncolin relationsalgebriquesentrevaleursdeefonctionsoudemfonctions |