Shidlovskii a donné une mesure d’indépendance linéaire de valeurs de -fonctions à coefficients de Taylor rationnels en un point rationnel qui n’est pas une singularité du système différentiel sous-jacent vérifié par ces -fonctions. Récemment, Beukers a prouvé un théorème d’indépendance linéaire qualitatif pour les valeurs en un point algébrique de -fonctions à coefficients de Taylor algébriques arbitraires. Dans cet article, nous obtenons un analogue de la mesure de Shidlovskii pour des valeurs de -fonctions arbitraires en des points algébriques. Cela nous permet de résoudre un problème longtemps ouvert : la valeur d’une -fonction en un point algébrique n’est jamais un nombre de Liouville. Nous prouvons également que des valeurs aux points rationnels de -fonctions à coefficients de Taylor rationnels sont linéairement indépendantes sur si et seulement si elles sont linéairement indépendantes sur . Nos méthodes reposent sur des améliorations de résultats obtenus par André et Beukers concernant la théorie des -opérateurs.
Shidlovskii has given a linear independence measure of values of -functions with rational Taylor coefficients at a rational point, not a singularity of the underlying differential system satisfied by these -functions. Recently, Beukers has proved a qualitative linear independence theorem for the values at an algebraic point of -functions with arbitrary algebraic Taylor coefficients. In this paper, we obtain an analogue of Shidlovskii’s measure for values of arbitrary -functions at algebraic points. This enables us to solve a long standing problem by proving that the value of an -function at an algebraic point is never a Liouville number. We also prove that values at rational points of -functions with rational Taylor coefficients are linearly independent over if and only if they are linearly independent over . Our methods rest upon improvements of results obtained by André and Beukers in the theory of -operators.
CC-BY 4.0
@article{JEP_2024__11__1_0,
author = {St\'ephane Fischler and Tanguy Rivoal},
title = {Values of $E$-functions are not {Liouville~numbers}},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1--18},
publisher = {\'Ecole polytechnique},
volume = {11},
year = {2024},
doi = {10.5802/jep.249},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.249/}
}
TY - JOUR AU - Stéphane Fischler AU - Tanguy Rivoal TI - Values of $E$-functions are not Liouville numbers JO - Journal de l’École polytechnique — Mathématiques PY - 2024 SP - 1 EP - 18 VL - 11 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.249/ DO - 10.5802/jep.249 LA - en ID - JEP_2024__11__1_0 ER -
%0 Journal Article %A Stéphane Fischler %A Tanguy Rivoal %T Values of $E$-functions are not Liouville numbers %J Journal de l’École polytechnique — Mathématiques %D 2024 %P 1-18 %V 11 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.249/ %R 10.5802/jep.249 %G en %F JEP_2024__11__1_0
Stéphane Fischler; Tanguy Rivoal. Values of $E$-functions are not Liouville numbers. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1-18. doi : 10.5802/jep.249. https://jep.centre-mersenne.org/articles/10.5802/jep.249/
[1] - “Diophantine properties of real numbers generated by finite automata”, Compositio Math. 142 (2006) no. 6, p. 1351-1372 | DOI | MR | Zbl
[2] - “Méthode de Mahler: relations linéaires, transcendance et applications aux nombres automatiques”, Proc. London Math. Soc. (3) 115 (2017) no. 1, p. 55-90 | DOI | Zbl
[3] - -functions and geometry, Aspects of Math., vol. E13, Friedr. Vieweg & Sohn, Braunschweig, 1989 | DOI
[4] - “Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité”, Ann. of Math. (2) 151 (2000) no. 2, p. 705-740 | DOI | Zbl
[5] - “Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence”, Ann. Sci. École Norm. Sup. (4) 47 (2014) no. 2, p. 449-467 | DOI | MR | Zbl
[6] - Transcendental number theory, Cambridge Math. Library, Cambridge University Press, Cambridge, 1990
[7] - “Diophantine approximation of Mahler numbers”, Proc. London Math. Soc. (3) 110 (2015) no. 5, p. 1157-1206 | DOI | MR | Zbl
[8] - “Équations différentielles linéaires et majorations de multiplicités”, Ann. Sci. École Norm. Sup. (4) 18 (1985) no. 1, p. 181-192 | DOI | Numdam | Zbl
[9] - “Effective estimates for global relations on Euler-type series”, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004) no. 2, p. 241-260 | DOI | Numdam | MR | Zbl
[10] - “A refined version of the Siegel-Shidlovskii theorem”, Ann. of Math. (2) 163 (2006) no. 1, p. 369-379 | DOI | MR | Zbl
[11] - “Minimization of differential equations and algebraic values of -functions”, 2022 | arXiv
[12] - Elements of mathematics. Algebra II. Chapters 4–7, Springer, Berlin, 2003
[13] - Approximation by algebraic numbers, Cambridge Tracts in Math., vol. 160, Cambridge University Press, Cambridge, 2004 | DOI
[14] - “Transcendental numbers”, in Number theory, IV, Encyclopaedia Math. Sci., vol. 44, Springer, Berlin, 1998, p. 1-345 | MR
[15] - “On the values of -functions”, Comment. Math. Helv. 89 (2014) no. 2, p. 313-341 | DOI | Zbl
[16] - “Arithmetic theory of -operators”, J. Éc. polytech. Math. 3 (2016), p. 31-65 | DOI | Numdam | MR | Zbl
[17] - “Microsolutions of differential operators and values of arithmetic Gevrey series”, Amer. J. Math. 140 (2018) no. 2, p. 317-348 | DOI | MR | Zbl
[18] - “Zur Approximation von ”, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 9 (1966), p. 3-14 | MR
[19] - “The André-Chudnovsky-Katz theorem in the broad sense”, North-West. Eur. J. Math. 7 (2021), p. 83-149 | MR | Zbl
[20] - “Zur Approximation der Exponentialfunktion und des Logarithmus. I”, J. reine angew. Math. 166 (1931), p. 118-136 | MR | Zbl
[21] - “Fields and Galois theory”, 2022, version 5.10, available at www.jmilne.org/math, 144 pages
[22] - “On the linear independence of values of -functions”, Mat. Sb. 187 (1996) no. 8, p. 93-108 | DOI | MR
[23] - “Valeurs algébriques de -fonctions aux points algébriques” | HAL
[24] - Transcendental numbers, De Gruyter Studies in Math., vol. 12, Walter de Gruyter & Co., Berlin, 1989 | DOI | MR
[25] - “Über einige Anwendungen diophantischer Approximationen.”, Abh. Preuß. Akad. Wiss., Phys.-Math. Kl. 1929 (1929) no. 1 | Zbl
[26] - “On rational approximations of values of a certain class of entire functions”, Mat. Sb. 186 (1995) no. 4, p. 555-590 | DOI | Zbl
Cité par Sources :