We show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the strong Lefschetz property. This is the first example of an explicit dual form with these properties.

For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition.

Moreover, we determine the Waring rank, the cactus rank, the resolution and the strong Lefschetz property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables.

We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

我们证明完全对称多项式是满足强 Lefschetz 属性的压缩 artinian Gorenstein 代数的对偶生成器。这是具有这些属性的显式对偶形式的第一个示例。

对于任意数量的变量中任意程度的完全对称形式，我们通过建立显式幂和分解来提供 Waring 等级的上限。

此外，我们确定了由对称三次形式定义的任何 Gorenstein 代数的 Waring 等级、仙人掌等级、分辨率和强 Lefschetz 属性。特别是，我们表明，通过增加变量的数量，可以使对称立方形式的 Waring 等级和仙人掌等级之间的差异任意大。

我们为四度和五度的通用对称形式的 Waring 等级提供上限。