Vol. 16, No. 1, 2022

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

Mats Boij, Juan Migliore, Rosa M. Miró-Roig and Uwe Nagel

Vol. 16 (2022), No. 1, 155–178
Abstract

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.

Keywords
Waring rank, cactus rank, symmetric forms, strong Lefschetz property, Macaulay duality, minimal free resolution, power sum decomposition, Gorenstein algebra
Mathematical Subject Classification
Primary: 13B25
Milestones
Received: 9 October 2020
Revised: 25 January 2021
Accepted: 6 May 2021
Published: 22 February 2022
Authors
Mats Boij
Department of Mathematics
KTH Royal Institute of Technology
Stockholm
Sweden
Juan Migliore
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States
Rosa M. Miró-Roig
Facultat de Mathemátiques i Informàtica
Universitat de Barcelona
Barcelona
Spain
Uwe Nagel
Department of Mathematics
University of Kentucky
Lexington, KY
United States