Explicit Bounds for Linear Forms in the Exponentials of Algebraic Numbers

TL;DR

This paper provides explicit bounds for linear forms in exponentials of algebraic numbers, combining effective Lindemann–Weierstrass theorem with algebraic counting methods.

Contribution

It introduces explicit lower bounds for such linear forms and establishes upper bounds based on algebraic number counting, advancing effective transcendence measures.

Findings

Explicit lower bounds for linear forms in exponentials of algebraic numbers.

Existence of linear forms with explicit upper bounds.

Application of constructive algebraic methods to transcendence theory.

Abstract

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.