  We study three variations of the Waring problem for polynomials, concerning
the Waring rank, the border rank and the cactus rank of a form and we show how
the Lefschetz properties of the associated algebra affect them. The main tool
is the theory of mixed Hessian matrix. We construct new families of wild forms,
that is, forms whose cactus rank, of schematic nature, is bigger then the
border rank, defined geometrically.
