Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers

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

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