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

TL;DR

This paper introduces a new explicit transcendence measure for exponential values at algebraic points, improving previous bounds by optimizing parameters in Siegel's determinant method applied to Hermite-Padé approximants.

Contribution

It provides a novel explicit exponent that enhances existing transcendence measures for exponential functions at algebraic arguments, especially for degrees greater than or equal to 2.

Findings

Improved transcendence measure for $e^eta$ at algebraic $eta$

Enhanced bounds for polynomial degrees $ ext{deg}(P) extgreater 1$

Recovery of previous bounds for linear polynomials ($ ext{deg}(P)=1$)

Abstract

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $\delta\ge 1$ and usual height $H\ge 1$, and let $\alpha\in \overline{\mathbb Q}^*$ 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-Pad{\'e} approximants to powers of the exponential function.