Related papers: On real typical ranks

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an…

Following the breakthrough of Croot, Lev, and Pach, Tao introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which…

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix…

The set ${\cal P}^{m\times n}_{r,d}$ of $m \times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \dots , \min \{m, n\}-1$, we show that ${\cal…

For a given projective variety $X$, the high rank loci are the closures of the sets of points whose $X$-rank is higher than the generic one. We show examples of strict inclusion between two consecutive high rank loci. Our first example is…

The paper establishes that the rank of a regular polygonal complex in 3-space E^3 cannot exceed 4, and that the only regular polygonal complexes of rank 4 in 3-space are the eight regular 4-apeirotopes.

The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We…

A new approach to the algebraic classification of second order symmetric tensors in 5-dimensional space-times is presented. The possible Segre types for a symmetric two-tensor are found. A set of canonical forms for each Segre type is…

Let $X(\RR)$ be a geometrically connected variety defined over $\RR$ and such that the set of all its (also complex) points $X(\CC)$ is non-degenerate. We introduce the notion of \emph{admissible rank} of a point $P$ with respect to $X$ to…

Motivated by a problem in computational complexity, we consider the behavior of rank functions for tensors and polynomial maps under random coordinate restrictions. We show that, for a broad class of rank functions called natural rank…

Over fields of characteristic unequal to $2$, we can identify symmetric matrices with homogeneous polynomials of degree $2$. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper,…

A vector space A of matrices is called rank-critical if any vector space that properly contains A has a strictly higher generic rank. I present a sufficient condition for A to be rank-critical, and apply this condition to prove that certain…

Let $\operatorname{r_{max}}(n,d)$ be the maximum Waring rank for the set of all homogeneous polynomials of degree $d>0$ in $n$ indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when…

Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of a great number of brane cosmological models is that in which the matter fields are confined on a…

We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that [6(n+1)/7] linear inequalities suffice to describe a convex n-gon up to a linear projection.

Let $p$ be an odd prime, $m\in {\mathbb N}$ and set $q=p^m$, $G=\operatorname{PSL}_n(q)$. Let $\theta$ be a standard graph automorphism of $G$, $d$ be a diagonal automorphism and $\operatorname{Fr}_q$ be the Frobenius endomorphism of…

Let $X(\mathbb {C})\subset \mathbb {P}^r(\mathbb {C})$ be an integral non-degenerate variety defined over $\mathbb {R}$. For any $q\in \mathbb {P}^r(\mathbb {R})$ we study the existence of $S\subset X(\mathbb {C})$ with small cardinality,…

Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we…

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and…

There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this…