# The set of forms with bounded strength is not closed

@article{Ballico2022TheSO,
title={The set of forms with bounded strength is not closed},
author={Edoardo Ballico and Arthur Bik and A. Oneto and Emanuele Ventura},
journal={Comptes Rendus. Math{\'e}matique},
year={2022}
}
• Published 2 December 2020
• Mathematics
• Comptes Rendus. Mathématique
The strength of a homogeneous polynomial (or form) $f$ is the smallest length of an additive decomposition expressing it whose summands are reducible forms. We show that the set of forms with bounded strength is not always Zariski-closed. In particular, if the ground field has characteristic $0$, we prove that the set of quartics with strength $\leq3$ is not Zariski-closed for a large number of variables.
7 Citations
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ArXiv
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A re-nement of Kumar's recent quadratic algebraic branching program size lower bound proof method (CCC 2017) is provided and examples in which the reﬁned method gives a better lower bound than the original one are shown.
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Vietnam journal of mathematics
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A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the
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We prove that the Buchweitz-Greueul-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the
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. We revisit Schmidt’s theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also ﬁnd a sharper
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• 2021
Abstract. We prove that strength and slice rank of homogeneous polynomials of degree d ≥ 5 over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a
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Linear and Multilinear Algebra
• 2021
A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible
Corrigendum to “Strength conditions, small subalgebras, and Stillman bounds in degree $\leq 4$”
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Transactions of the American Mathematical Society
• 2020
. The statement and proof of a proposition, which appeared in Trans. Amer. Math. Soc. 373 (2020), no. 7, 4757–4806, about the locus where strength of a form is at most k are corrected: the locus is

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