Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers

In this paper, we study linear forms $\lambda = \beta_1{\mathrm{e}}^{\alpha_1}+\cdots+\beta_m{\mathrm{e}}^{\alpha_m}$, where $\alpha_i$ and $\beta_i$ are algebraic numbers. An explicit lower bound for $|\lambda|$ is proved, which is derived from "th\'eor\`eme de Lindemann--Weierstrass effectif" via constructive methods in algebraic computation. Besides, an explicit upper bound for the minimal $|\lambda|$ is established on systematic results of counting algebraic numbers.