Degree-free bounds for dependence relations

Let a1,…, an be non-zero albegraic numbers and let l1(a1),…ln(an) denote arbitrary fixed values of the logarithms of a1,…n, respectively Given that l1(a1),…ln(an) are linearly dependent over Q, the existence of non-trival dependence relation between these numbers with integer coefficients of low absolute values can be proved. Existing results of this kind give bounds for the absolute values of the coefficients which are expressions in the degree D = [Q(a1…an): Q], the heights of a1,…an and the magnitudes of the logarithms involved.