  We give upper-bounds for the dimension of some linear systems. The theorem
improves the differential Horace method introduced by Alexander-Hirschowitz,
and was conjectured by Simpson. Possible applications are the calculus of the
dimension of linear systems of hypersurfaces in a projective space $\PP^n$ with
generically prescribed singularities, and the calculus of collisions of fat
points in $\PP^2$. These applications will be treated independently but a
simple example in the introduction explains how the theorem will be used.
