DIOPHANTINE APPROXIMATION AND ALGEBRAIC I NDEPENDENCE OF LOGARITHMS

We prove that any transcendental complex number is well approximated b y algebraic numbers of large degree and bounded absolute logarithmic h eight. Next we extend this result to a statement on simultaneous dioph antine approximation for any finite subset of a field of transcendence degree 1 over Q. This tool enables us to introduce a new method for a lgebraic independence, which we develop in the context of several para meters subgroups of linear algebraic groups. We show for instance that if log alpha(1),...,log alpha(n) are Q-linearly independent logarithm s of algebraic numbers in a field of transcendence degree 1 over Q, th en for any non zero quadratic form Q is an element of Q[X-1,...,X-n], the number Q(log alpha(1),...,log alpha(n)) does not vanish.