Linear equations in variables which lie in a multiplicative group

J. -H. Evertse; H. P. Schlickewei; W. M. Schmidt

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Linear equations in variables which lie in a multiplicative group

J. -H. Evertse·H. P. Schlickewei·W. M. Schmidt

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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 $(K^\ast)^n$ of finite rank r. Given $A_2,...,a_n\in K^\ast$ write $A(a_1,...,a_n,Γ)$ for the number of solutions x=(x_1,...,x_n)\in Γ$ of the equation a_1x_1+...+a_nx_n=1$, such that no proper subsum of $a_1x_1+...+a_nx_n$ vanishes. We derive an explicit upper bound for $A(a_1,...,a_n,Γ)$ which depends only on the dimension n and on the rank r.

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BibTeX RIS

J. -H. Evertse, H. P. Schlickewei, W. M. Schmidt. 2004-09-30. Linear equations in variables which lie in a multiplicative group. https://arxiv.org/abs/math/0409604

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