Sufficient conditions for Strassen’s additivity conjecture

Teitler, Zach

doi:10.1215/ijm/1488186021

Cited by 13 publications

(17 citation statements)

References 21 publications

(41 reference statements)

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“…The latter bound was lightly improved in [173] (Theorem 2.3). Formula (26) can be generalized to higher order differentials.…”

Section: Remark 14 Notice That If Deg( ∩ Z) Is Exactly D + 1 + K Thmentioning

confidence: 98%

“…In [187], the authors proved that Strassen's conjecture holds whenever the summands are in either one or two variables. In [171,173], the authors provided conditions on the summands to guarantee that additivity of the symmetric-ranks holds. For example, in [173], the author showed that whenever the catalecticant bound (27) (or the lower bound given by Theorem 29) is sharp for all the F i 's, then Strassen's conjecture holds, and the corresponding bound for ∑ s i=1 F i is also sharp.…”

Section: Remark 14 Notice That If Deg( ∩ Z) Is Exactly D + 1 + K Thmentioning

confidence: 99%

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Decomposability of Tensors

2019

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“…The latter bound was lightly improved in [173] (Theorem 2.3). Formula (26) can be generalized to higher order differentials.…”

Section: Remark 14 Notice That If Deg( ∩ Z) Is Exactly D + 1 + K Thmentioning

confidence: 98%

Section: Remark 14 Notice That If Deg( ∩ Z) Is Exactly D + 1 + K Thmentioning

confidence: 99%

Decomposability of Tensors

2019

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“…Conjecture 2 and its analogues have been proven when either T 1 or T 2 has dimension at most two, when rankpT 1 q can be determined by the so called substitution method [LM17], when dimpV 1 q " 2 both for the rank and the border rank [BGL13], when T 1 , T 2 are symmetric that is homogeneous polynomials in disjoint sets of variables, either T 1 , T 2 is a power, or both T 1 and T 2 have two variables, or either T 1 or T 2 has small rank [CCC15], and also for other classes of homogeneous polynomials [CCO17], [Tei15].…”

Section: Strassen's Conjecturementioning

confidence: 99%

On Comon’s and Strassen’s Conjectures

2018

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Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

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“…A homogeneous polynomial

f

is called a direct sum if, after a change of variables, it can be written as a sum of two or more polynomials in disjoint sets of variables:

f = f_{1} false(x_{1}, \dots, x_{a} false) + f_{2} false(x_{a + 1}, \dots, x_{n} false) .

Homogeneous direct sums are the subject of a well‐known symmetric Strassen's additivity conjecture postulating that the Waring rank of

f

in is the sum of the Waring ranks of

f_{1}

and

f_{2}

(see, for example, ). Direct sums also play a special role in the study of geometric invariant theory (GIT) stability of associated forms .…”

Section: Introductionmentioning

confidence: 99%

“…Homogeneous direct sums are the subject of a well-known symmetric Strassen's additivity conjecture postulating that the Waring rank of f in (1.1) is the sum of the Waring ranks of f 1 and f 2 (see, for example, [15]). Direct sums also play a special role in the study of geometric invariant theory (GIT) stability of associated forms [10].…”

Section: Introductionmentioning

confidence: 99%

Direct sum decomposability of polynomials and factorization of associated forms

Fedorchuk

2020

Proc. London Math. Soc.

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We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non‐zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if‐and‐only‐if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic 0 or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.

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Sufficient conditions for Strassen’s additivity conjecture

Cited by 13 publications

References 21 publications

Decomposability of Tensors

Decomposability of Tensors

On Comon’s and Strassen’s Conjectures

Direct sum decomposability of polynomials and factorization of associated forms

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