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29 March 2024
 
  » arxiv » math.NT/0409604

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Linear equations in variables which lie in a multiplicative group
J.-H. Evertse ; H. P. Schlickewei ; W. M. Schmidt ;
Date 30 Sep 2004
Journal Ann. of Math. (2), Vol. 155 (2002), no. 3, 807--836
Subject Number Theory | math.NT
AbstractLet 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 $A_2,...,a_nin K^ast$ 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.
Source arXiv, math.NT/0409604
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