  We study the obstruction degrees of translates of sub-tori of multiplicative
tori and we show how they are connected to lower bounds for the essential
minimum of these varieties. In particular, we combine our computations with
results of A. Schinzel and of F. Amoroso - R. Dvornicich, that we thus extend
to translates of sub-tori defined over CM fields. This shows that the essential
minimum of these varieties have a geometric behavior similar to that of
varieties which are not translates of sub-tori. We also refine and extend to
translates of sub-tori a lower bound obtained by D. Bertrand in connexion with
relations of multiplicative dependence of algebraic points.
  Finaly, we prove that in the functional case the conjectural lower bound for
the essential minimum of sub-varieties of multiplicative tori is true and its
proof is elementary.
