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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 |
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Vol. 16 (2022), No. 1, 155–178
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Abstract |
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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
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Mathematical Subject Classification
Primary: 13B25
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Milestones
Received: 9 October 2020
Revised: 25 January 2021
Accepted: 6 May 2021
Published: 22 February 2022
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Authors |
| Mats Boij | |
| Department of Mathematics KTH Royal Institute of Technology Stockholm Sweden |
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| Juan Migliore | |
| Department of Mathematics University of Notre Dame Notre Dame, IN United States |
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| Rosa M. Miró-Roig | |
| Facultat de Mathemátiques i
Informàtica Universitat de Barcelona Barcelona Spain |
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| Uwe Nagel | |
| Department of Mathematics University of Kentucky Lexington, KY United States |
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