Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers

Abstract

In this paper, we study linear forms \[λ= β_1\mathrm{e}^{α_1}+\cdots+β_m\mathrm{e}^{α_m},\] where ${\alpha }_i$ and ${\beta }_i$ are algebraic numbers. An explicit lower bound for the absolute value of $\lambda$ is proved, which is derived from "théorème de Lindemann--Weierstrass effectif" via constructive methods in algebraic computation. Besides, the existence of $\lambda$ with an explicit upper bound is established on the result of counting algebraic numbers.