Related papers: Interactions of zeros of of polynomials and multip…

In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if ${\mathbb A}=(A_{1},...,A_{n})$ is a $n$-tuple of positive definite…

Let ${\bf M}_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over an arbitrary field $\mathbb{F}$. We consider linear maps $\Phi: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ preserving matrices annihilated by a fixed…

Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p…

The lattice cell in the ${i+1}^{st}$ row and ${j+1}^{st}$ column of the positive quadrant of the plane is denoted $(i,j)$. If $\mu$ is a partition of $n+1$, we denote by $\mu/ij$ the diagram obtained by removing the cell $(i,j)$ from the…

Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x n matrices over {0,1,2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of…

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix…

The number of Monotone Triangles with bottom row k1 < k2 < ... < kn is given by a polynomial alpha(n; k1,...,kn) in n variables. The evaluation of this polynomial at weakly decreasing sequences k1 >= k2 >= ... >= kn turns out to be…

We give necessary and sufficient conditions, in the form of matrix identities, for a polynomial f in C[X,Y] to be a component of a polynomial automorphism of C^2 and to be a component of a Keller polynomial mapping of C^2, respectively…

Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…

We study the problem of determining a matrix whose $k$th multiplicative compound is a prescribed matrix~$M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals~$M$ is characterized in terms of $\rank(M)$. On…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of…

Let $p$ be a prime. Suppose that integers $r$, $e$, $d$ such that $r \ge 2$, $e \ge 0$, $0 \le d \le p$ are given. Let $f(x)=s_0 x^r + s_1 x^{r-1} + \cdots + s_r$ be a generic polynomial of degree $r$ in characteristic $p$. We put…

An n\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell} M_{\ell,k} = 0. Dietzfelbinger, Hromkovi\v{c}, and Schnitger (1996) showed that n \le (\rk…

A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0, P_\pm]$ $=$ $\pm P_\pm$, $[P_+, P_-]$ $=$ $\phi^{(m)}(P_0)$ where $\phi^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ with the…

Given a matrix with partitions of its rows and columns and entries from a field, we give the necessary and sufficient conditions that it has a non--singular submatrix with certain number of rows from each row partition and certain number of…

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which…

Linear statistics, a random variable build out of the sum of the evaluation of functions at the eigenvalues of a N times N random matrix,sum[j=1 to N]f(xj) or tr f(M), is an ubiquitous statistical characteristics in random matrix theory.…