Abelian Tensors

  title={Abelian Tensors},
  author={Joseph Landsberg and Mateusz Michałek},

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.

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a

Bounds on complexity of matrix multiplication away from CW tensors

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.



Typical Tensorial Rank

The typical rank R(f) of a format f is the rank of Zariski almost all tensors of that format. Following Strassen [505] and Lickteig [331] we determine the asymptotic growth of the function R and

Stratification of the fourth secant variety of Veronese varieties via the symmetric rank

Abstract. If is a projective non-degenerate variety, the X-rank of a point is defined to be the minimum integer r such that P belongs to the span of r points of X. We describe the complete

Toward a Salmon Conjecture

Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3×3×4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9, furnishing a computational solution of an open problem in algebraic statistics.

On the Ranks and Border Ranks of Symmetric Tensors

Improved lower bounds for the rank of a symmetric tensor are provided by considering the singularities of the hypersurface defined by the polynomial.

A note on the gap between rank and border rank

Varieties of reductions for $gl\_n$

We study the varieties of reductions associated to the variety of rank one matrices in $\fgl\_n$. These varieties are defined as natural compactifications of the different ways to write the identity

Generalizations of Strassen's Equations for Secant Varieties of Segre Varieties

We define many new examples of modules of equations for secant varieties of Segre varieties that generalize Strassen's commutation equations (Strassen, 1988). Our modules of equations are obtained by

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n