Related papers: Polynomial Maps with Constants on Matrix Algebra

We continue the analysis in [H. Osaka and J. Tomiyama, Double piling structure of matrix monotone functions and of matrix convex functions, Linear and its Applications 431(2009), 1825 - 1832] in which the followings three assertions at each…

We construct, for every even dimensional sphere $S^n$, $n >1$, and every odd integer $k$, a homogeneous polynomial map $f: S^{n}\to S^{n}$ of Brouwer degree $k$ and algebraic degree $2|k|-1$.

We consider manifolds whose transition maps are restrictions of polynomial mappings $\mathbb{R}^n\to\mathbb{R}^n$, and use them to give an equivalent statement of the Jacobian conjecture over the real field.

Let $K$ be any field and $x = (x_1,x_2,\ldots,x_n)$. We classify all matrices $M \in {\rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${\rm rk} M \le 2$. As a special case, we describe all such matrices…

To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when…

Given a family $\{ f_{\lambda} \}_{\lambda \in \Lambda}$ of polynomial maps of degree $d$ where $\Lambda$ is the set of parameters, a polynomial map $f_{\lambda_0}$ is called {\it $J$-stable in $\Lambda$} if there exists a neighborhood of…

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The…

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$,…

Let M be a manifold endowed with a symmetric affine connection $\Gamma.$ The aim of this paper is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T^{*} M and the space of second-order…

Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper…

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…

We prove the complete monotonicity on $(0,\infty)^n$ for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result…

A combinatorial structure, $\mathcal{F}$, with counting sequence $\{a_n\}_{n\ge 0}$ and ordinary generating function $G_\mathcal{F}=\sum_{n\ge0} a_n x^n$, is positive algebraic if $G_\mathcal{F}$ satisfies a polynomial equation…

We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…

If a real analytic nonexpansive map on a polyhedral normed space has a nonempty fixed point set, then we show that there is an isometry from an affine subspace onto the fixed point set. As a corollary, we prove that for any real analytic…

Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero. In the paper we give a criterion of nearly irreducibility for a given polynomial f in…

We prove that if $F$ is a Lipschitz map from the set of all complex $n\times n$ matrices into itself with $F(0)=0$ such that given any $x$ and $y$ we have that $% F\left( x\right) -F\left( y\right) $ and $x-y$ have at least one common…

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra $M_n(K)$ over a field $K$ endowed with its canonical $\mathbb{Z}_n$-grading (Vasilovsky's grading). We…

A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…

We classify all two-dimensional simple algebras (which may be non-associative) over an algebraically closed field. For each two-dimensional algebra $\mathcal{A}$, we describe a minimal (with respect to inclusion) generating set for the…