Let $P\in \mathbb{Z}[X]\setminus \lbrace 0\rbrace $ be of degree $\delta \ge 1$ and usual height $H\ge 1$, and let $\alpha \in \overline{\mathbb{Q}}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^\alpha $: for any $\varepsilon >0$, there exists $c>0$ such that $\vert P(e^\alpha )\vert >c/H^{\mu (d,\delta )+\varepsilon }$ where the exponent $\mu (d,\delta )=(4d^2-2d)\delta +2d-1$. Zheng obtained a better result in 1991 with $\mu (d,\delta )=(4d^2-2d)\delta -1$. In this paper, we provide a new explicit exponent $\mu (d,\delta )$ which improves on Zheng’s transcendence measure for all $\delta \ge 2$ and all $d\ge 2$. When $\delta =1$, we recover his bound for all $d\ge 2$, which had in fact already been obtained by Kappe in 1966. Our improvement rests upon the optimization of an accessory parameter in Siegel’s classical determinant method applied to Hermite–Padé approximants to powers of the exponential function.
Soit $P\in \mathbb{Z}[X]\setminus \lbrace 0\rbrace $ de degré $\delta \ge 1$ et de hauteur $H\ge 1$, et soit $\alpha \in \overline{\mathbb{Q}}^*$ de degré $d\ge 2$. Mahler a obtenu en 1931 la mesure de transcendance suivante pour $e^\alpha $ : pour tout $\varepsilon >0$, il existe $c>0$ tel que $\vert P(e^\alpha )\vert >c/H^{\mu (d,\delta )+\varepsilon }$, où l’exposant $\mu (d,\delta )=(4d^2-2d)\delta +2d-1$. Zheng a obtenu une meilleure mesure en 1991 avec $\mu (d,\delta )=(4d^2-2d)\delta -1$. Dans cet article, nous obtenons un nouvel exposant $\mu (d,\delta )$ qui améliore la mesure de transcendance de Zheng pour tout $\delta \ge 2$ et tout $d\ge 2$. Lorsque $\delta =1$, nous retrouvons sa mesure pour tout $d\ge 2$, mesure qui avait en fait déjà été obtenue par Kappe en 1966. Notre amélioration repose sur l’optimisation d’un paramètre accessoire dans la méthode classique du déterminant de Siegel, appliquée aux approximants de Hermite–Padé de puissances de la fonction exponentielle.
Keywords: Exponential function, Transcendence measure, Hermite–Padé approximants
Stéphane Fischler 1 ; Tanguy Rivoal 2
CC-BY-ND 4.0
Stéphane Fischler; Tanguy Rivoal. A new transcendence measure for the values of the exponential function at algebraic arguments. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2026), pp. 7-19. doi: 10.5802/pmb.67
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author = {St\'ephane Fischler and Tanguy Rivoal},
title = {A new transcendence measure for the values of the exponential function at algebraic arguments},
journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
pages = {7--19},
year = {2026},
publisher = {Presses universitaires de Franche-Comt\'e},
doi = {10.5802/pmb.67},
language = {en},
url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.67/}
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