Let $V$ be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of $V$ of height bounded by invariants associated to any variety containing $V$ . In particular, we determine whether such a set is or is not dense in $V$ . We then prove that these sets can always be written as the intersection of $V$ with a finite union of translates of tori of which we control the sum of the degrees.
¶ As a consequence, we prove a conjecture by Amoroso and David up to a logarithmic factor