Comptes Rendus
Research article - Number theory
Relations algébriques entre valeurs de E-fonctions ou de M-fonctions
[Algebraic relations between values of Siegel E-functions and Mahler M-functions]
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1215-1241.

We show that all algebraic relations over ¯ 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 ¯(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.

Nous montrons que toutes les relations algébriques sur ¯ entre les valeurs prises par des E-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 ¯(z) entre les fonctions considérées. Nous obtenons un résultat analogue pour les M q -fonctions de Mahler, dans lequel les relations dites σ q -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 E-fonctions et celle des M q -fonctions.

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DOI: 10.5802/crmath.634
Classification: 11J81, 11J91
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

1 Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Relations alg\'ebriques entre valeurs de $E$-fonctions ou de $M$-fonctions},
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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/

[1] Boris Adamczewski; Jason P. Bell A problem about Mahler functions, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 4, pp. 1301-1355 | DOI | MR | Zbl

[2] Boris Adamczewski; Jason P. Bell; Daniel Smertnig A height gap theorem for coefficients of Mahler functions, J. Eur. Math. Soc., Volume 25 (2023) no. 7, pp. 2525-2571 | DOI | MR | Zbl

[3] Boris Adamczewski; Thomas Dreyfus; Charlotte Hardouin Hypertranscendence and linear difference equations, J. Am. Math. Soc., Volume 34 (2021) no. 2, pp. 475-503 | DOI | MR | Zbl

[4] Boris Adamczewski; Thomas Dreyfus; Charlotte Hardouin; Michael Wibmer Algebraic independence and linear difference equations, J. Eur. Math. Soc., Volume 26 (2024) no. 5, pp. 1899-1932 | DOI | MR | Zbl

[5] Boris Adamczewski; Colin Faverjon Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques, Proc. Lond. Math. Soc., Volume 115 (2017) no. 1, pp. 55-90 | DOI | MR | Zbl

[6] Boris Adamczewski; Colin Faverjon Méthode de Mahler, transcendance et relations linéaires : aspects effectifs, J. Théor. Nombres Bordeaux, Volume 30 (2018) no. 2, pp. 557-573 | DOI | Numdam | MR | Zbl

[7] Boris Adamczewski; Colin Faverjon Mahler’s method in several variables and finite automata (2020) (to appear in Ann. Math., https://arxiv.org/abs/2012.08283)

[8] Boris Adamczewski; Colin Faverjon A new proof of Nishioka’s theorem in Mahler’s method, C. R. Math. Acad. Sci. Paris, Volume 361 (2023), pp. 1011-1028 | DOI | MR | Zbl

[9] Yves André Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, Ann. Math., Volume 151 (2000) no. 2, pp. 705-740 | DOI | MR | Zbl

[10] Yves André Séries Gevrey de type arithmétique. II. Transcendance sans transcendance, Ann. Math., Volume 151 (2000) no. 2, pp. 741-756 | DOI | MR | Zbl

[11] Yves André Solution algebras of differential equations and quasi-homogeneous varieties : a new differential Galois correspondence, Ann. Sci. Éc. Norm. Supér., Volume 47 (2014) no. 2, pp. 449-467 | DOI | MR | Zbl

[12] Boris Adamczewski; Tanguy Rivoal Exceptional values of E-functions at algebraic points, Bull. Lond. Math. Soc., Volume 50 (2018) no. 4, pp. 697-708 | DOI | MR | Zbl

[13] Jean-Paul Allouche; Jeffrey Shallit Automatic sequences. Theory, applications, generalizations, Cambridge University Press, 2003, xvi+571 pages | DOI | MR | Zbl

[14] Alin Bostan; Frédéric Chyzak; Marc Giusti; Romain Lebreton; Grégoire Lecerf; Bruno Salvy; Éric Schost Algorithmes Efficaces en Calcul Formel (2018) https://hal.science/aecf/ (686 pp.)

[15] Jason P. Bell; Michael Coons; Eric Rowland The rational-transcendental dichotomy of Mahler functions, J. Integer Seq., Volume 16 (2013) no. 2, 13.2.10, 11 pages | MR | Zbl

[16] Paul-Georg Becker k-regular power series and Mahler-type functional equations, J. Number Theory, Volume 49 (1994) no. 3, pp. 269-286 | DOI | MR | Zbl

[17] Frits Beukers A refined version of the Siegel–Shidlovskii theorem, Ann. Math., Volume 163 (2006) no. 1, pp. 369-379 | DOI | MR | Zbl

[18] Nicolas Bourbaki Elements of mathematics. Algebra II. Chapters 4–7, Springer, 2003, viii+461 pages | DOI | MR | Zbl

[19] Alin Bostan; Tanguy Rivoal; Bruno Salvy Minimization of differential equations and algebraic values of E-functions, Math. Comput., Volume 93 (2024) no. 347, pp. 1427-1472 | DOI | MR | Zbl

[20] Thomas Becker; Volker Weispfenning Gröbner bases : a computational approach to commutative algebra. In cooperation with Heinz Kredel, Graduate Texts in Mathematics, 141, Springer, 1993 | DOI | Zbl

[21] Ruyong Feng Hrushovski’s algorithm for computing the Galois group of a linear differential equation, Adv. Appl. Math., Volume 65 (2015), pp. 1-37 | DOI | MR | Zbl

[22] Ruyong Feng On the computation of the Galois group of linear difference equations, Math. Comput., Volume 87 (2018) no. 310, pp. 941-965 | DOI | MR | Zbl

[23] Stéphane Fischler; Tanguy Rivoal Effective algebraic independence of values of E-functions, Math. Z., Volume 305 (2023) no. 3, 48, 17 pages | DOI | MR | Zbl

[24] Stéphane Fischler; Tanguy Rivoal Values of E-functions are not Liouville numbers, J. Éc. Polytech., Math., Volume 11 (2024), pp. 1-18 | DOI | MR | Zbl

[25] Javier Fresán Une introduction aux périodes, Périodes et transcendance, Les Éditions de l’École polytechnique, 2022

[26] Maxim Kontsevich; Don Zagier Periods, Springer (2001), pp. 771-808 | DOI

[27] Serge Lang Algebra, Graduate Texts in Mathematics, 211, Springer, 2002, xvi+914 pages | DOI | MR | Zbl

[28] Kumiko Nishioka New approach in Mahler’s method, J. Reine Angew. Math., Volume 407 (1990), pp. 202-219 | DOI | MR | Zbl

[29] Levente Nagy; Tamás Szamuely A general theory of André’s solution algebras, Ann. Inst. Fourier, Volume 70 (2020) no. 5, pp. 2103-2129 | DOI | Numdam | MR | Zbl

[30] Patrice Philippon Groupes de Galois et nombres automatiques, J. Lond. Math. Soc., Volume 92 (2015) no. 3, pp. 596-614 | DOI | MR | Zbl

[31] Tanguy Rivoal Les E-fonctions et G-fonctions de Siegel, Périodes et transcendance, Les Éditions de l’École polytechnique, 2022

[32] Julien Roques On the algebraic relations between Mahler functions, Trans. Am. Math. Soc., Volume 370 (2018) no. 1, pp. 321-355 | DOI | MR | Zbl

[33] Andrei Borisovich Shidlovskii Transcendental numbers, De Gruyter Studies in Mathematics, 12, Walter de Gruyter, 1989, xx+466 pages | DOI | MR | Zbl

[34] Marius van der Put; Michael F. Singer Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, 2003, xviii+438 pages | DOI | MR | Zbl

[35] Marius van der Put; Michael F. Singer Galois theory of difference equations, Lecture Notes in Mathematics, 1666, Springer, 1997, viii+180 pages | DOI | MR | Zbl

[36] Michel Waldschmidt Transcendence of periods : the state of the art, Pure Appl. Math. Q., Volume 2 (2006) no. 2, pp. 435-463 (Special Issue : In honor of John H. Coates. Part 2) | DOI | MR | Zbl

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