Abelian Tensors

@article{Landsberg2015AbelianT,
  title={Abelian Tensors},
  author={Joseph Landsberg and Mateusz Michałek},
  journal={ArXiv},
  year={2015},
  volume={abs/1504.03732}
}

Irreversibility of Structure Tensors of Modules

It is shown that it is impossible to prove ω = 2 by starting with structure tensors of modules of fixed degree and using arbitrary restrictions, and it implies that the same is impossible byStarting with 1A-generic non-diagonal tensor of fixed size with minimal border rank.

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Three families of minimal border rank tensors are presented: they come from highest weight vectors, smoothable algebras, or monomial algebraes; they also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating.

Hermitian K-theory via oriented Gorenstein algebras

We show that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along finite Gorenstein morphisms with trivialized dualizing sheaf. As an application, we

A geometric study of Strassen's asymptotic rank conjecture and its variants

Surprisingly, this dimension equals the dimension of the set of oblique tensors, a less restrictive class of tensors that Strassen identified as useful for his laser method.

Geometric conditions for strict submultiplicativity of rank and border rank

The X -rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p . This notion is naturally submultiplicative under tensor product.

On Comon’s and Strassen’s Conjectures

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

On Degeneration of Tensors and Algebras

This work describes the smoothable algebra associated to the Coppersmith-Winograd tensor and proves a lower bound for the border rank of the tensor used in the "easy construction" of Coppermith and Winograd.

Towards a geometric approach to Strassen’s asymptotic rank conjecture

A first geometric study of three varieties in C m, including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry, to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptic rank of any tight tensor is minimal.

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