Linear equations in variables which lie in a multiplicative group

2004/09/30 by J. -H. Evertse, H. P. Schlickewei, W. M. Schmidt
Mathematics · #math.NT

paper · pdf

published as Ann. of Math. (2), Vol. 155 (2002), no. 3, 807--836

arxiv created 2004/09/30 · arxiv updated 2009/12/01

Abstract

Let K be a field of characteristic 0 and let n be a natural number. Let Gamma be a subgroup of the multiplicative group (K^∗)ⁿ of finite rank r. Given A₂,...,a_n∈ K^∗ write A(a₁,...,a_n,Γ) for the number of solutions x=(x₁,...,x_n)∈ Γ of the equation a₁x₁+...+a_nx_n=1, such that no proper subsum of a₁x₁+...+a_nx_n vanishes. We derive an explicit upper bound for A(a₁,...,a_n,Γ) which depends only on the dimension n and on the rank r.

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