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

Abstract

Let P ∈ ℤ [ X ] ∖ { 0 } be of degree δ ≥ 1 and usual height H ≥ 1 , and let α ∈ ℚ ¯ * be of degree d ≥ 2 . Mahler proved in 1931 the following transcendence measure for e α : for any ε > 0 , there exists c > 0 such that | P ( e α ) | > c / H μ ( d , δ ) + ε where the exponent μ ( d , δ ) = ( 4 d 2 - 2 d ) δ + 2 d - 1 . Zheng obtained a better result in 1991 with μ ( d , δ ) = ( 4 d 2 - 2 d ) δ - 1 . In this paper, we provide a new explicit exponent μ ( d , δ ) which improves on Zheng’s transcendence measure for all δ ≥ 2 and all d ≥ 2 . When δ = 1 , we recover his bound for all d ≥ 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.