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

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

doi:10.2140/ant.2022.16.155

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

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

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	155-178
Number of pages	24
Journal	Algebra and Number Theory
Volume	16
Issue number	1
DOIs	https://doi.org/10.2140/ant.2022.16.155
State	Published - 2022

Bibliographical note

Funding Information:
The authors want to thank CIRM in Trento, MFO in Oberwolfach and the University of Kentucky for support and hospitality. They also want to thank the referee for useful questions and suggestions that helped improve the exposition. Migliore was partially supported by a Simons Foundation grant #309556. Miró-Roig was partially supported by MTM2016-78623-P. Nagel was partially supported by Simons Foundation grants #317096 and #636513. MSC2020: 13B25. Keywords: Waring rank, cactus rank, symmetric forms, strong Lefschetz property, Macaulay duality, minimal free resolution, power sum decomposition, Gorenstein algebra.

Funding Information:
The authors want to thank CIRM in Trento, MFO in Oberwolfach and the University of Kentucky for support and hospitality. They also want to thank the referee for useful questions and suggestions that helped improve the exposition. Migliore was partially supported by a Simons Foundation grant #309556. Mir?-Roig was partially supported by MTM2016-78623-P. Nagel was partially supported by Simons Foundation grants #317096 and #636513.

Publisher Copyright:
© 2022, Mathematical Science Publishers. All rights reserved.

Keywords

Cactus rank
Gorenstein algebra
Macaulay duality
Minimal free resolution
Power sum decomposition
Strong Lefschetz property
Symmetric forms
Waring rank

ASJC Scopus subject areas

Algebra and Number Theory

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10.2140/ant.2022.16.155

Cite this

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title = "Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms",

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 = "Cactus rank, Gorenstein algebra, Macaulay duality, Minimal free resolution, Power sum decomposition, Strong Lefschetz property, Symmetric forms, Waring rank",

author = "Mats Boij and Juan Migliore and Mir{\'o}-Roig, {Rosa M.} and Uwe Nagel",

note = "Funding Information: The authors want to thank CIRM in Trento, MFO in Oberwolfach and the University of Kentucky for support and hospitality. They also want to thank the referee for useful questions and suggestions that helped improve the exposition. Migliore was partially supported by a Simons Foundation grant #309556. Mir{\'o}-Roig was partially supported by MTM2016-78623-P. Nagel was partially supported by Simons Foundation grants #317096 and #636513. MSC2020: 13B25. Keywords: Waring rank, cactus rank, symmetric forms, strong Lefschetz property, Macaulay duality, minimal free resolution, power sum decomposition, Gorenstein algebra. Funding Information: The authors want to thank CIRM in Trento, MFO in Oberwolfach and the University of Kentucky for support and hospitality. They also want to thank the referee for useful questions and suggestions that helped improve the exposition. Migliore was partially supported by a Simons Foundation grant #309556. Mir?-Roig was partially supported by MTM2016-78623-P. Nagel was partially supported by Simons Foundation grants #317096 and #636513. Publisher Copyright: {\textcopyright} 2022, Mathematical Science Publishers. All rights reserved.",

year = "2022",

doi = "10.2140/ant.2022.16.155",

language = "English",

volume = "16",

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AU - Miró-Roig, Rosa M.

AU - Nagel, Uwe

N1 - Funding Information: The authors want to thank CIRM in Trento, MFO in Oberwolfach and the University of Kentucky for support and hospitality. They also want to thank the referee for useful questions and suggestions that helped improve the exposition. Migliore was partially supported by a Simons Foundation grant #309556. Miró-Roig was partially supported by MTM2016-78623-P. Nagel was partially supported by Simons Foundation grants #317096 and #636513. MSC2020: 13B25. Keywords: Waring rank, cactus rank, symmetric forms, strong Lefschetz property, Macaulay duality, minimal free resolution, power sum decomposition, Gorenstein algebra. Funding Information: The authors want to thank CIRM in Trento, MFO in Oberwolfach and the University of Kentucky for support and hospitality. They also want to thank the referee for useful questions and suggestions that helped improve the exposition. Migliore was partially supported by a Simons Foundation grant #309556. Mir?-Roig was partially supported by MTM2016-78623-P. Nagel was partially supported by Simons Foundation grants #317096 and #636513. Publisher Copyright: © 2022, Mathematical Science Publishers. All rights reserved.

PY - 2022

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N2 - 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.

AB - 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.

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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