## 34 Citations

### Irreversibility of Structure Tensors of Modules

- Computer ScienceCollectanea Mathematica
- 2022

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

- MathematicsMathematics
- 2018

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

- MathematicsArXiv
- 2021

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

- Mathematics
- 2021

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

- MathematicsArXiv
- 2018

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

- Mathematics
- 2019

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

- MathematicsMathematics
- 2018

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

- Mathematics, Computer ScienceMFCS
- 2016

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

- MathematicsCollectanea Mathematica
- 2020

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.

## References

SHOWING 1-10 OF 57 REFERENCES

### Typical Tensorial Rank

- Mathematics
- 1997

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

- Mathematics
- 2013

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

- MathematicsExp. Math.
- 2011

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

- Mathematics, Computer ScienceFound. Comput. Math.
- 2010

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

### Varieties of reductions for $gl\_n$

- Mathematics
- 2004

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

- Mathematics
- 2006

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

- Mathematics
- 1992

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…