Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers

Abstract

In this paper, we study linear forms λ=β1eα1+...βmeαm, 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 "theoreme me 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.