Explicit Bounds for Linear Forms in the Exponentials of Algebraic\n Numbers

2021/12/09 by Cheng-Chao Huang, Huang, Cheng-Chao
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Numerical Methods and Algorithms #Polynomial and algebraic computation #Symbolic Computation (cs.SC)

paper · pdf · doi:10.48550/arxiv.2112.05004

openalex publication_date 2021/12/09 · openalex created_date 2022/11/04 · openalex updated_date 2026/07/28

Abstract

In this paper, we study linear forms \
lambda =\n
beta₁
mathrme
^alpha₁+
cdots+
beta_m
mathrme
^alpha_m, where\n\α_i and \β_i are algebraic numbers. An explicit lower bound for the\nabsolute value of \λ is proved, which is derived from "th 'eor `eme de\nLindemann--Weierstrass effectif" via constructive methods in algebraic\ncomputation. Besides, the existence of \λ with an explicit upper bound\nis established on the result of counting algebraic numbers.\n

Related

• A note on linear forms in two logarithms: the argument of an algebraic power
• An explicit Baker type lower bound of exponential values
• On the approximation of the values of exponential function and logarithm by algebraic numbers
• Minorations simultanées de formes linéaires de logarithmes de nombres algébriques
• A kit for linear forms in three logarithms