A new transcendence measure for the values of the exponential function at algebraic arguments
Stéphane Fischler, Tanguy Rivoal
TL;DR
The paper develops a new explicit transcendence measure for $e^{\alpha}$ with $\alpha$ algebraic, introducing the exponent $\psi(d,\delta,\lambda)$ and optimizing an accessory parameter to improve on prior bounds. By combining explicit Hermite–Padé approximants to powers of the exponential with Siegel’s determinant method, it reduces the problem to bounding a determinant and extracting a lower bound for linear forms in $e^{\alpha}$. The main result yields a bound $|P(e^{\alpha})| > c H^{-\psi(d,\delta,\lambda)-\varepsilon}$ with $\lambda$ chosen to minimize $\psi$, improving Zheng’s bound for $\delta\ge 2$ and recovering Kappe’s result when $\delta=1$. The paper also provides a completely explicit version with fully specified constants and proves auxiliary monotonicity and inequality properties of the exponent, along with a broader construction that, while general, does not surpass the main bounds.
Abstract
Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^α$: for any $\varepsilon\>0$, there exists $c\>0$ such that $\vert P(e^α)\vert\>c/H^{μ(d,δ)+\varepsilon}$ where the exponent $μ(d,δ)=(4d^2-2d)δ+2d-1$. Zheng obtained a better result in 1991 with $μ(d,δ)=(4d^2-2d)δ-1$. In this paper, we provide a new explicit exponent $μ(d,δ)$ which improves on Zheng's transcendence measure for all $δ\ge 2$ and all $d\ge 2$. When $δ=1$, 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.