2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150379We 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.17 pages, in french, also available at http://193.49.162.129/~evain/home.htmlAlgebraic Geometrytext