no. G4
Géométrie algébrique
The set of forms with bounded strength is not closed
Ballico, Edoardo1 ; Bik, Arthur2  ; Oneto, Alessandro1 ; Ventura, Emanuele3
1 Università di Trento, Via Sommarive, 14 - 38123 Povo (Trento), Italy
2 MPI for Mathematics in the Sciences, Leipzig, Germany
3 Politecnico di Torino, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 371-380.

The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not always Zariski-closed. More specifically, if the ground field is algebraically closed, we prove that the set of quartics with strength ≤3 is not Zariski-closed for a large number of variables.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.302
Classification : 15A21, 13A02, 14R20
Ballico, Edoardo 1 ; Bik, Arthur 2 ; Oneto, Alessandro 1 ; Ventura, Emanuele 3

1 Università di Trento, Via Sommarive, 14 - 38123 Povo (Trento), Italy
2 MPI for Mathematics in the Sciences, Leipzig, Germany
3 Politecnico di Torino, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 
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     title = {The set of forms with bounded strength is not closed},
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     pages = {371--380},
     publisher = {Acad\'emie des sciences, Paris},
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Ballico, Edoardo; Bik, Arthur; Oneto, Alessandro; Ventura, Emanuele. The set of forms with bounded strength is not closed. Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 371-380. doi : 10.5802/crmath.302. http://archive.numdam.org/articles/10.5802/crmath.302/

[1] Ananyan, Tigran; Hochster, Melvin Small subalgebras of polynomial rings and Stillman’s conjecture, J. Am. Math. Soc., Volume 33 (2020) no. 1, pp. 291-309 | DOI | MR | Zbl

[2] Ananyan, Tigran; Hochster, Melvin Strength conditions, small subalgebras, and Stillman bounds in degree ≤4, Trans. Am. Math. Soc. (2020) no. 373, pp. 4757-4806 | DOI | MR | Zbl

[3] Ballico, Edoardo; Bik, Arthur; Oneto, Alessandro; Ventura, Emanuele Strength and slice rank of forms are generically equal (2021) (https://arxiv.org/abs/2102.11549, to appear in Isr. J. Math.)

[4] Ballico, Edoardo; Ventura, Emanuele The strength for line bundles, Math. Scand., Volume 127 (2021) no. 3, pp. 405-440

[5] Bik, Arthur Strength and Noetherianity for infinite Tensors, Ph. D. Thesis, Universität Bern (2020) https://biblio.unibe.ch/download/eldiss/20bik_ma.pdf

[6] Bik, Arthur; Draisma, Jan; Eggermont, Rob H. Polynomials and tensors of bounded strength, Commun. Contemp. Math., Volume 21 (2019) no. 07, 1850062, 24 pages | MR | Zbl

[7] Bik, Arthur; Draisma, Jan; Eggermont, Rob H.; Snowden, Andrew The geometry of polynomial representations (2021) (https://arxiv.org/abs/2105.12621)

[8] Bik, Arthur; Oneto, Alessandro On the strength of general polynomials (2021) (https://doi.org/10.1080/03081087.2021.1947955, to appear in Linear Multilinear Algebra) | DOI

[9] Derksen, Harm; Eggermont, Rob H.; Snowden, Andrew Topological Noetherianity for cubic polynomials, Algebra Number Theory, Volume 11 (2017) no. 9, pp. 2197-2212 | DOI | MR | Zbl

[10] Draisma, Jan Topological Noetherianity of polynomial functors, J. Am. Math. Soc., Volume 32 (2019) no. 3, pp. 691-707 | DOI | MR | Zbl

[11] Erman, Daniel; Sam, Steven V.; Snowden, Andrew Big polynomial rings and Stillman’s conjecture, Invent. Math., Volume 218 (2019) no. 2, pp. 413-439 | DOI | MR | Zbl

[12] Erman, Daniel; Sam, Steven V.; Snowden, Andrew Big Polynomial Rings with Imperfect Coefficient Fields, Mich. Math. J., Volume 70 (2021) no. 3, pp. 649-672 | MR | Zbl

[13] Harris, Joe Algebraic geometry: A first course, Graduate Texts in Mathematics, 133, Springer, 1992 | DOI | Zbl

[14] Kazhdan, David; Polishchuk, Alexander Linear subspaces of minimal codimension in hypersurfaces (2021) (https://arxiv.org/abs/2107.08080)

[15] Kazhdan, David; Polishchuk, Alexander Schmidt rank and singularities (2021) (https://arxiv.org/abs/2104.10198)

[16] Kazhdan, David; Polishchuk, Alexander Schmidt rank of quartics over perfect fields (2021) (https://arxiv.org/abs/2110.10244)

[17] Kazhdan, David; Ziegler, Tamar On ranks of polynomials, Algebr. Represent. Theory, Volume 21 (2018) no. 5, pp. 1017-1021 | DOI | MR | Zbl

[18] Lang, Serge Hilbert’s Nullstellensatz in infinite-dimensional space, Proc. Am. Math. Soc., Volume 3 (1952), pp. 407-410 | MR | Zbl

[19] Schmidt, Wolfgang M. The density of integer points on homogeneous varieties, Acta Math., Volume 154 (1985) no. 3-4, pp. 243-296 | DOI | MR | Zbl

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