Related papers: On the typical rank of real binary forms

Fix integers $m \ge 2$, $s\ge 5$ and $d\ge 2s+2$. Here we describe the possible symmetric tensor ranks $\le 2d+s-7$ of all symmetric tensors (or homogeneous degree $d$ polynomials) in $m+1$ variables with border rank $s$.

In this paper, we study the real and the complex Waring rank of reducible cubic forms. In particular, we compute the complex rank of all reducible cubic forms. In the real case, for all reducible cubics, we either compute or bound the real…

The Ciliberto-Di Gennaro conjecture predicts that a nodal hypersurface of degree $d\geq 3$ with at most $2(d-2)(d-1)$ nodes is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has…

The aim of this paper is the introduction of a new method for the numerical computation of the rank of a three-way array. We show that the rank of a three-way array over R is intimately related to the real solution set of a system of…

Previous research established that the maximal rank of the abstract regular polytopes whose automorphism group is a transitive proper subgroup of $\mbox{S}_n$ is $n/2 + 1$. Up to isomorphism and duality, when $n\geq 12$, there are only two…

We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we…

Let K be a field of characteristic different from 2 and let V be a vector space of dimension n over K. Let M be a non-zero subspace of symmetric bilinear forms defined on V x V and let r=rank(M) denote the set of different positive integers…

We classify the four dimensional perfect non-simple evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

Let $G$ be a digraph and $r(G)$ be its rank. Many interesting results on the rank of an undirected graph appear in the literature, but not much information about the rank of a digraph is available. In this article, we study the rank of a…

In this note, we generalize the technique developed in [13] and prove that every 5xn matrix of tropical rank at most 3 has Kapranov rank at most 3, for the ground field that contains at least 4 elements. For the ground field either F_2 or…

We introduce the notion of ''maximal rank type'' for representations of quivers, which requires certain collections of maps involved in the representation to be of maximal rank. We show that real root representations of quivers are of…

We introduce a new foundation rank based in the relation of dividing between partial types. We call DU to this rank. We also introduce a new way to define the D rank over formulas as a foundation rank. In this way, SU, DU and D are…

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly…

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly…

Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$.…

All quasi-affine connected Generalized Dynkin Diagram with rank $= 5$ are found. All quasi-affine Nichols (Lie braided) algebras with rank $ 5$ are also found.

A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the…

For a fixed rational number g, not equal to -1,0 or 1 and integers a and d we consider the set of primes p for which the order of g(mod p) is congruent to a(mod d). For d=4 and d=3 it is shown that, under the Generalized Riemann Hypothesis,…

We are interested by holomorphic $d$-webs $W$ of codimension one in a complex $n$-dimensional manifold $M$. If they are ordinary, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled), we proved in [CL]…

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the…