MICHEL WALDSCHMIDT
A well-known conjecture states that linearly independent logarithms of algebraic numbers are algebraically independent over the field of rational numbers. So far, it is not yet known that there exist two algebraically independent logarithms of algebraic numbers. On the other hand, D. Roy has shown that the above conjecture is equivalent to a conjectural description of the rank of matrices whose entries are either algebraic numbers or else logarithms of algebraic numbers. From this point of view, half of the conjecture is known: the actual rank of such a matrix is at least half the conjectural rank.
We consider a similar question for commutative algebraic groups. We show a connection with a density problem, and we prove a partial result by means of the theorem of the algebraic subgroup.