Related papers: On Maximum, Typical and Generic Ranks

Let $G$ be a special $p$-group. If $G$ is of rank two, or $G$ is of maximum rank with $|G^p|\leq p$, then we describe the complex irreducible projective representations of $G$.

Given a differential or $q$-difference equation $P$ of order $n$, we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of $P$ plus $n$. We also prove…

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible…

It is well known that a best rank-$R$ approximation of order-3 tensors may not exist for $R\ge 2$. A best rank-$(R,R,R)$ approximation always exists, however, and is also a best rank-$R$ approximation when it has rank (at most) $R$. For…

We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree…

We prove that the probability $P_N$ for a real random Gaussian $N\times N\times 2$ tensor to be of real rank $N$ is $P_N=(\Gamma((N+1)/2))^N/G(N+1)$, where $\Gamma(x)$, $G(x)$ denote the gamma and Barnes $G$-functions respectively. This is…

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension.

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph $G$ is said to be maximal if any reduced…

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…

The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard…

We show that the sum of ranks of two matrix polynomials is the same as the sum of the rank of the matrix obtained by applying the greatest common divisor of the polynomials, with the rank of the matrix obtained by applying the lowest common…

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…

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…

The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in…

In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point…

The Waring rank of the generic $d \times d$ determinant is bounded above by $d \cdot d!$. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum…

The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a…

Given a nonnegative matrix M with rational entries, we consider two quantities: the usual positive semidefinite (psd) rank, where the matrix is factored through the cone of real symmetric psd matrices, and the rational-restricted psd rank,…

At the time of writing, the general problem of finding the maximal Waring rank for homogeneous polynomials of fixed degree and number of variables (or, equivalently, the maximal symmetric rank for symmetric tensors of fixed order and in…

In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for m\times n\times p tensors over R in the following cases: (1) 3\leq m\leq \rho(n) and (m-1)(n-1)+1\leq p\leq (m-1)n, where \rho\ is the…