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 the absolute value of $\lambda$ is proved, which is derived from "th\'eor\`eme 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.