Related papers: Border Ranks of Monomials
We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and…
We present new lower bounds for the symmetric border rank of the n x n determinant for all n. Further lower bounds are given for the 3 x 3 permanent.
We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of Landsberg and Teitler and generalize it to a lower bound for rank with respect to arbitrary varieties, improving the bound given by…
For odd n, I write down tensors in C^n\otimes C^n\otimes C^n of border rank 2n-1, showing the non-triviality of the Young-flattening equations of Landsberg-Ottaviani. I also study the border rank of the tensors of Alexeev et. al., showing…
Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric…
We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds…
We prove border rank bounds for a class of $GL(V)$-invariant tensors in $V^*\otimes U\otimes W$, where $U$ and $W$ are $GL(V)$-modules. These tensors correspond to spaces of matrices of constant rank. In particular we prove lower bounds for…
We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound $2.431$ on $\omega$. We also…
We exhibit, for each even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives the lower bound…
We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C^4 x C^4 -> C^4. This…
Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an…
The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds…
Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way…
We study the relationship between the commutative and the non-commutative rank of a linear matrix. We give examples that show that the ratio of the two ranks comes arbitrarily close to 2. Such examples can be used for giving lower bounds…
There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…
We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…
We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the…
Motivated by connections between algebraic complexity lower bounds and tensor decompositions, we investigate Koszul-Young flattenings, which are the main ingredient in recent lower bounds for matrix multiplication. Based on this tool we…
An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast…
We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. Our technique simultaneously yields a dynamical…