Apolarity, border rank and multigraded Hilbert scheme

Buczyńska, Weronika; Buczyński, Jarosław

doi:10.48550/arxiv.1910.01944

Cited by 7 publications

(36 citation statements)

References 28 publications

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“…Lie's Theorem and the Normal Form Lemma allow us to take I 111 to be B T -fixed. The Fixed Ideal Theorem of [BB20] uses the same reasoning to generalize this to prove B T -invariance for all multigraded components I ijk , not just a finite number of multigraded components.…”

Section: Methodsmentioning

confidence: 99%

“…Border Apolarity. In order to establish larger lower bounds on R(T sl 3 ) than can be achieved by Koszul flattenings and border substitution for T sl 3 , we will use border apolarity, as developed in [BB20] and [CHL19].…”

Section: Methodsmentioning

confidence: 99%

“…I ijk will exist, since the Grassmannian is compact, however, the resulting ideal I may not be saturated. See [BB20] for further discussion on this.…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…In [BB20], see Theorems 3.15 (Border Apolarity) for proof of the first part and see Theorem 4.3 (Fixed Ideal Theorem) for a proof of the second part. Theorem 3.15 and 4.3 of [BB20] are not stated here as they are stated in greater generality than we require using the language of schemes. We remark that the Weak Border Apolarity Theorem provides sufficiency that if a border rank r decomposition exists, then there will exist another border rank r decomposition satisfying the given conditions.…”

Section: Methodsmentioning

confidence: 99%

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On the structure tensor of $\mathfrak{sl}_n$

Bari¹

2021

Preprint

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The structure tensor of sl n , denoted T sln , is the tensor arising from the Lie bracket bilinear operation on the set of traceless n × n matrices over C. This tensor is intimately related to the well studied matrix multiplication tensor. Studying the structure tensor of sl n may provide further insight into the complexity of matrix multiplication and the "hay in a haystack" problem of finding explicit sequences tensors with high rank or border rank. We aim to find new bounds on the rank and border rank of this structure tensor in the case of sl 3 and sl 4 . We additionally provide bounds in the case of the lie algebras so 4 and so 5 . The lower bounds on the border ranks were obtained via various recent techniques, namely Koszul flattenings, border substitution, and border apolarity. Upper bounds on the rank of T sl3 are obtained via numerical methods that allowed us to find an explicit rank decomposition.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…I ijk will exist, since the Grassmannian is compact, however, the resulting ideal I may not be saturated. See [BB20] for further discussion on this.…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

See 3 more Smart Citations

On the structure tensor of $\mathfrak{sl}_n$

Bari¹

2021

Preprint

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New lower bounds for matrix multiplication and the 3x3 determinant

Conner,

Harper,

Landsberg

2019

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Let M ⟨u,v,w⟩ ∈ C uv ⊗C vw ⊗C wu denote the matrix multiplication tensor (and write M ⟨n⟩ = M ⟨n,n,n⟩ ) and let det3 ∈ (C 9 ) ⊗3 denote the determinant polynomial considered as a tensor. For a tensor T , let R(T ) denote its border rank. We (i) give the first hand-checkable algebraic proof that R(M ⟨2⟩ ) = 7, (ii) prove R(M ⟨223⟩ ) = 10, and R(M ⟨233⟩ ) = 14, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was M ⟨2⟩ , (iii) prove R(M ⟨3⟩ ) ≥ 17, (iv) prove R(det3) = 17, improving the previous lower bound of 12, (v) prove R(M ⟨2nn⟩ ) ≥ n 2 + 1.32n for all n ≥ 25 (previously only R(M ⟨2nn⟩ ) ≥ n 2 + 1 was known) as well as lower bounds for 4 ≤ n ≤ 25, and (vi) prove R(M ⟨3nn⟩ ) ≥ n 2 + 2n for all n ≥ 21, where previously only R(M ⟨3nn⟩ ) ≥ n 2 + 2 was known, as well as lower bounds for 4 ≤ n ≤ 21,.Our results utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to tensors with symmetry to obtain an algorithm that, given a tensor T with a large symmetry group and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all potential border rank r decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. The two key ingredients are: (i) the use of a multi-graded ideal associated to a border rank r decomposition of any tensor, and (ii) the exploitation of the large symmetry group of T to restrict to B T -invariant ideals, where B T is a maximal solvable subgroup of the symmetry group of T .

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Vanishing Hessian, wild forms and their border VSP

Huang

Michałek

Ventura

2019

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We show that forms with vanishing Hessian and of minimal border rank are wild, i.e. their smoothable rank is strictly larger than their border rank. This discrepancy is caused by the difference between the limit of spans of a family of zerodimensional schemes and the span of their flat limit. We exhibit an infinite series of wild forms of every degree d ≥ 3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers VSP of these forms in the corresponding multigraded Hilbert scheme.

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Apolarity, border rank and multigraded Hilbert scheme

Cited by 7 publications

References 28 publications

On the structure tensor of $\mathfrak{sl}_n$

On the structure tensor of $\mathfrak{sl}_n$

New lower bounds for matrix multiplication and the 3x3 determinant

Vanishing Hessian, wild forms and their border VSP

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