Waring problems and the Lefschetz properties

We study three variations of the Waring problem for homogeneous polynomials, concerning the Waring rank, the border rank and the cactus rank of a form. We show how the Lefschetz properties of the associated algebra affect them. We construct new families of wild forms, that is, forms whose cactus rank, of schematic nature, is bigger than the border rank, defined geometrically.