Related papers: The Waring Problem for Matrix Algebras

Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in $n$ variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic…

The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$ has degree…

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions,…

A double star is a tree with two internal vertices. It is known that the Gy\'arf\'as-Sumner conjecture holds for double stars, that is, for every double star $H$, there is a function $f$ such that if $G$ does not contain $H$ as an induced…

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…

To cater to the needs of (Zero Knowledge) proofs for (mathematical) proofs, we describe a method to transform formal sentences in 2x2-matrices over multivariate polynomials with integer coefficients, such that usual proof-steps like…

A noncommutative (nc) polynomial is called (globally) trace-positive if its evaluation at any tuple of operators in a tracial von Neumann algebra has nonnegative trace. Such polynomials emerge as trace inequalities in several matrix or…

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…

The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank…

We present an example of a subfield $\mathcal{F}\subset\mathbb{R}$ and a matrix $A$ whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, $A$ can be…

The present paper considers extremal combinatorics questions in the language of matrices. An $s$-matrix is a matrix with entries in $\{0,1,\ldots, s-1\}$. An $s$-matrix is simple if it has no repeated columns. A matrix $F$ is a…

Viewing a bivariate polynomial f in R[x,t] as a family of univariate polynomials in t parametrized by real numbers x, we call f real rooted if this family consists of monic polynomials with only real roots. If f is the characteristic…

Let $n$ and $k$ be integers such that $1\le k\le n$ and $f(x)$ be a nonzero polynomial of integer coefficients such that $f(m)\ne 0$ for any positive integer $m$. For any $k$-tuple $\vec{s}=(s_1, ..., s_k)$ of positive integers, we define…

In the Nonnegative Matrix Factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$ where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r…

We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.

Let ${\cal A}=\{A_1,\ldots, A_r\}$ be a partition of a set $\{1,\ldots,m\}\times\{1,\ldots, n\}$ into $r$ nonempty subsets, and $A=(a_{ij})$ be an $m\times n$ matrix. We say that $A$ has a pattern ${\cal A}$ provided that $a_{ij}=a_{i'j'}$…

The noncommutative (or mixed) trace algebra $T_{nd}$ is generated by $d$ generic $n\times n$ matrices and by the algebra $C_{nd}$ generated by all traces of products of generic matrices, $n,d\geq 2$. It is known that over a field of…

We show that there does not exist a generalised polynomial which vanishes precisely on the set of powers of two. In fact, if $k \geq 2$ is and integer and $g \colon \mathbb{N} \to \mathbb{R}$ is a generalised polynomial such that $g(k^n) =…

A square matrix of order n with $n\geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…