A new transcendence measure for the values of the exponential function at algebraic arguments

Abstract

Let $P\in \mathbb{Z}[X]\setminus \{0\}$ be of degree $\delta \ge 1$ and usual height $H\ge 1$, and let $\alpha \in {\overline{\mathbb{Q}}}^{\ast }$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^{\alpha }$: for any $\epsilon \>0$, there exists $c\>0$ such that $|P(e^{\alpha })|\>c/H^{\mu (d,\delta )+\epsilon }$ 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-Pad{é} approximants to powers of the exponential function.