Linear Equations in Variables which Lie in a Multiplicative Group

2002/05/01 by J.-H. Evertse, Jan‐Hendrik Evertse, H. P. Schlickewei +3 · 28 citations
Mathematics · #Algebraic Geometry and Number Theory #Finite Group Theory Research #Advanced Algebra and Geometry

paper · doi:10.2307/3062133

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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