Vanishing Hessian, wild forms and their border VSP

Huang, Hang; Michałek, Mateusz; Ventura, Emanuele

doi:10.48550/arxiv.1912.13174

Cited by 2 publications

(2 citation statements)

References 22 publications

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“…However, the variant of apolarity for border rank has been stated only very recently (see [5]). Nevertheless, it has already been applied in [13] and [21]. In order to formulate it, we have to consider multigraded Hilbert schemes, see [19].…”

Section: Rank and Border Rank Of A Polynomialmentioning

confidence: 99%

Distinguishing secant from cactus varieties

Gałązka¹,

Mańdziuk²,

Rupniewski³

2020

Preprint

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In this paper we study homogeneous polynomials and vector subspaces of a polynomial ring that are divisible by a large power of a linear form. We compute their cactus and border cactus ranks. We show that for d ≥ 5, the component of the cactus variety κ14(ν d (P 6 )) other than the secant variety σ14(ν d (P 6 )) consists of degree d polynomials divisible by (d − 3)-th power of a linear form. For d ≥ 6 we present an algorithm for deciding whether a point in the cactus variety κ14(ν d (P 6 )) belongs to the secant variety σ14(ν d (P 6 )). Analogously, we show that for d ≥ 5, the component of the Grassmann cactus variety κ8,3(ν d (P 4 )) other than the Grassmann secant variety σ8,3(ν d (P 4 )) consists of subspaces divisible by (d − 2)-th power of a linear form. Finally, for d ≥ 5 we present an algorithm for deciding whether a point in the Grassmann cactus variety κ8,3(ν d (P 4 )) belongs to the Grassmann secant variety σ8,3(ν d (P 4 )).

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Section: Rank and Border Rank Of A Polynomialmentioning

confidence: 99%

Distinguishing secant from cactus varieties

Gałązka¹,

Mańdziuk²,

Rupniewski³

2020

Preprint

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“…Representation of polynomials by Hessian determinants has indeed proved to be a delicate topic: see [22] for a famous mistake by Hesse about his eponymous determinant. Hesse's mistake was about polynomials with vanishing Hessian, a topic that remains of interest to this day [28].…”

Section: Try To Find Constantsmentioning

confidence: 99%

Derandomization and absolute reconstruction for sums of powers of linear forms

Koiran¹,

Skomra²

2019

Preprint

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We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results, we give an algorithm for the following problem: given a homogeneous polynomial f (x 1 , . . . , x n ) of degree 3, decide whether it can be written as a sum of cubes of linearly independent linear forms with complex coefficients. Compared to previous algorithms for the same problem, the two main novel features of this algorithm are:(i) It is an algebraic algorithm, i.e., it performs only arithmetic operations and equality tests on the coefficients of the input polynomial f . In particular, it does not make any appeal to polynomial factorization.

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Vanishing Hessian, wild forms and their border VSP

Cited by 2 publications

References 22 publications

Distinguishing secant from cactus varieties

Distinguishing secant from cactus varieties

Derandomization and absolute reconstruction for sums of powers of linear forms

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