Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions

Adamczewski, Boris; Faverjon, Colin

Comptes Rendus. Mathématique, Tome 362 (2024) no. G10, pp. 1215-1241

Nous montrons que toutes les relations algébriques sur $\overline{\mathbb{Q}}$ 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 $\overline{\mathbb{Q}}\left(z\right)$ entre les fonctions considérées. Nous obtenons un résultat analogue pour les $M_q$-fonctions de Mahler, dans lequel les relations dites ${\sigma }_q$-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.

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 $\overline{\mathbb{Q}}\left(z\right)$ 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.

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

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

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

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

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

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

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

[11] André, Yves 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 | Numdam | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[36] Waldschmidt, Michel 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