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We study three variations of the Waring problem for polynomials, concerningthe Waring rank, the border rank and the cactus rank of a form and we show howthe Lefschetz properties of the associated algebra affect them. The main toolis 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 theborder rank, defined geometrically.

Waring problems and the Lefschetz properties