2025/02/25 by Stéphane Fischler, Fischler, Stéphane, Tanguy Rivoal +1
Mathematics · #FOS: Mathematics #Fuzzy Systems and Optimization #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2502.17992
openalex publication_date 2025/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let P∈ \mathbb Z[X]∖\0\ be of degree δ≥ 1 and usual height H≥ 1, and let α∈ \mathbb Q^* be of degree d≥ 2. Mahler proved in 1931 the following transcendence measure for eα: for any ε>0, there exists c>0 such that \vert P(eα)\vert>c/Hμ(d,δ)+ε where the exponent μ(d,δ)=(4d2-2d)δ+2d-1. Zheng obtained a better result in 1991 with μ(d,δ)=(4d2-2d)δ-1. In this paper, we provide a new explicit exponent μ(d,δ) which improves on Zheng's transcendence measure for all δ≥ 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.