Waring and cactus ranks and Strong Lefschetz Property for annihilators of symmetric forms

TL;DR

This paper investigates the algebraic properties of symmetric forms, establishing explicit dual generators satisfying the Strong Lefschetz Property, and provides bounds and exact values for Waring and cactus ranks for various symmetric forms.

Contribution

It introduces the first explicit dual form with the Strong Lefschetz Property for compressed Gorenstein algebras and determines ranks and properties of symmetric forms of specific degrees.

Findings

Complete symmetric polynomials are dual generators with Strong Lefschetz Property.

Explicit upper bounds for Waring rank of symmetric forms.

The difference between Waring and cactus rank can be arbitrarily large.

Abstract

In this note 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.