Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms

In this note we show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying Strong Lefschetz Property. This is first example an explicit form with these properties. For forms any degree in number variables, provide upper bound for Waring rank by establishing power sum decomposition. Moreover, determine rank, cactus resolution and Property algebra defined a cubic form. particular, difference between can be made arbitrarily large increasing variables. We bounds generic degrees four five.