# 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} }

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.

## 9 Citations

### Degree-restricted strength decompositions and algebraic branching programs

- Mathematics, Computer ScienceArXiv
- 2022

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.

### The Geometry of Polynomial Representations

- MathematicsInternational Mathematics Research Notices
- 2022

We define a GL-variety to be an (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial…

### Universality of High-Strength Tensors

- MathematicsVietnam journal of mathematics
- 2022

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…

### Matrix factorizations of generic polynomials

- Mathematics
- 2021

. We prove that the Buchweitz-Greuel-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…

### Schmidt rank and singularities

- Mathematics
- 2021

. 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…

### Strength and slice rank of forms are generically equal

- Mathematics
- 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…

### On the strength of general polynomials

- MathematicsLinear 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 $\leq 4$”

- MathematicsTransactions 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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