Related papers: Faber polynomials of matrices for non-convex sets

Paradigms of bilinear maps f between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of f with Keller C^n_c-maps (on suitable…

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result:…

The Amitsur-Levitski theorem asserts that $M_n(F)$ satisfies a polynomial identity of degree $2n$. (Here, $F$ is a field and $M_n(F)$ is the algebra of $n \times n$ matrices over $F$). It is easy to give examples of subalgebras of $M_n(F)$…

Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. We extend this characterization to all…

We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on…

This paper is devoted to the study of multigraded algebras and multigraded linear series. For an $\mathbb{N}^s$-graded algebra $A$, we define and study its volume function $F_A:\mathbb{N}_+^s\to \mathbb{R}$, which computes the asymptotics…

The motivating question for this work is a long standing open problem, posed by Nisan (1991), regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question…

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a non-composite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if $f$ is a…

For a function g(w) analytic and univalent in {w:1<|w|<\infty} with a simple pole at \infty and a continuous extension to {w:|w|\geq 1}, we consider the Faber polynomials F_n(z), n=0,1,2,..., associated to g(w) via their generating function…

Let us denote ${\cal V}$, the finite dimensional vector spaces of functions of the form $\psi(x) = p_n(x) + f(x) p_m(x)$ where $p_n(x)$ and $p_m(x)$ are arbitrary polynomials of degree at most $n$ and $m$ in the variable $x$ while $f(x)$…

Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in…

Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with…

Given a vector function ${\bf F}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in…

We show that there are functions $f$ in the H\"older class $C^{ { \alpha }}[0,1]$, $1< { \alpha }<2$ such that $f|_{A}$ is not convex, nor concave for any $A { \subset } [0,1]$ with $ { \bar { dim }_M } A> { \alpha }-1$. Our earlier result…

In this paper, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for certain matrix operators on the Fibonacci difference sequence spaces l_{p}(F) and l_{infinite}(F) to be compact, where…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

Let $X\subset\mathbb{R}^n$ be a convex closed and semialgebraic set and let $f$ be a polynomial positive on $X$. We prove that there exists an exponent $N\geq 1$, such that for any $\xi\in\mathbb{R}^n$ the function…

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…

The numerical range in the quaternionic setting is, in general, a non convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense,…