Linear Equations in Variables which Lie in a Multiplicative Group

2002/05/01 by J.-H. Evertse, Jan‐Hendrik Evertse, H. P. Schlickewei +3 · 225 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Finite Group Theory Research #Group (periodic table) #Lie group #Mathematical analysis #Mathematics #Multiplicative function #Pure mathematics

paper · doi:10.2307/3062133

published in Annals of Mathematics 155(3), 807 (Princeton University)

openalex publication_date 2002/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

Let K be a field of characteristic 0 and let n be a natural number. Let Γ be a subgroup of the multiplicative group (K∗)n of finite rank r. Given a1,..., an ∈ K∗ write A(a1,..., an,Γ) for the number of solutions x = (x1,..., xn) ∈ Γ of the equation a1x1 +... + anxn = 1, such that no proper subsum of a1x1 +... + anxn vanishes. We derive an explicit upper bound for A(a1,..., an,Γ) which depends only on the dimension n and on the rank r. 1

Citations

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