Linear equations in variables which lie in a multiplicative group

Details Linear equations in variables which lie in a multiplicative group Linear equations in variables which lie in a multiplicative group

2004-09-30

arxiv.org/abs/math/0409604v1

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

Mathematics

Number Theory

Scientific paper

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 $A(a_1,...,a_n,\Gamma)$ for the number of solutions x=(x_1,...,x_n)\in \Gamma$ 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 $A(a_1,...,a_n,\Gamma)$ which depends only on the dimension n and on the rank r.

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