On the maximal multiplicative complexity of a family of bilinear forms

Atkinson, M. D.; Stephens, N. M.

doi:10.1016/0024-3795(79)90026-0

Cited by 34 publications

(29 citation statements)

References 5 publications

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“…The bound cannot be improved in general. The structure of the proof of the theorem relies on an idea contained in [2], as will be clarified in Section 3.…”

Section: Resultsmentioning

confidence: 99%

“…It was shown in [2] that the statement is true when the ground field is the field of the complex numbers. The proof exploits in a crucial way both the separability properties and the algebraic closeness of C. The interest in Statement 1 is mainly justified by the following proposition.…”

Section: Generalizationsmentioning

confidence: 99%

“…A trivial upper bound is m 2 , as follows from Proposition 1 below. In [2], Atkinson and Stephens proved that the maximum rank over the field of complex numbers is bounded above by 1 2 m 2 + O(m) and, as far as we know, this is still the best result of its kind.…”

Section: Introductionmentioning

confidence: 98%

“…The bound cannot be improved in general. Note however that, as stated in [2], the maximum rank of a 3 × 3 × 3 -tensor over C is five. Finally, in Section 3, we discuss how our work relates to [2], and indicate a possible way to extend our results.…”

Section: Introductionmentioning

confidence: 99%

“…Elaborating on a technique introduced in [2], in Section 2 we prove that the rank of a 3×3×3 -tensor is at most six. The bound cannot be improved in general.…”

Section: Introductionmentioning

confidence: 99%

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On the rank of 3×3×3-tensors

Lavrauw

Pavan

Zanella

2013

Linear and Multilinear Algebra

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Abstract. Let U , V and W be finite dimensional vector spaces over the same field. The rank of a tensor τ in U ⊗ V ⊗ W is the minimum dimension of a subspace of U ⊗ V ⊗ W containing τ and spanned by fundamental tensors, i.e. tensors of the form u⊗v⊗w for some u in U , v in V and w in W . We prove that if U , V and W have dimension three, then the rank of a tensor in U ⊗ V ⊗ W is at most six, and such a bound cannot be improved in general. Moreover we discuss how the techniques employed in the proof might be extended to prove upper bounds for the rank of a tensor in U ⊗ V ⊗ W when the dimensions of U , V and W are higher.

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“…The bound cannot be improved in general. The structure of the proof of the theorem relies on an idea contained in [2], as will be clarified in Section 3.…”

Section: Resultsmentioning

confidence: 99%

Section: Generalizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

“…Elaborating on a technique introduced in [2], in Section 2 we prove that the rank of a 3×3×3 -tensor is at most six. The bound cannot be improved in general.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations