ON LINEAR INDEPENDENCE

It is the aim of the present work to prove, under appropriate conditio ns, lower estimates for the dimension of Ow1 + ... + Qw(m) over Q, whe re w1, ..., w(m) are given real numbers. In particular, if this dimens ion is m, i.e. if w1, ..., w(m) are linearly independent over Q, we ar e also interested in a quantitative version of this fact. Our qualitat ive theorems generalize a result of Nesterenko. Its formulation is qui te similar to the ''axiomatization'' of methods for algebraic independ ence, as it became usual during the last decade.