Related papers: On Multilinear Polynomials In Four Variables Evalu…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…

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…

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by…

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 $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in a algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\times 3$…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector…

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$,…

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…

Kaplansky asked about the possible images of a polynomial $f$ in several noncommuting variables. In this paper we consider the case of $f$ a Lie polynomial. We describe all the possible images of $f$ in $M_2(K)$ and provide an example of…

Let ${\mathbb F}$ be an arbitrary field. We consider a commutative, non-associative, $4$-dimensional algebra ${\mathfrak M}$ of the rock, the paper and the scissors with unit over ${\mathbb F}$ and we prove that the image over ${\mathfrak…

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…

In this paper we prove the generalized Kaplansky conjecture for the Jordan algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

Let $F$ be an algebraically closed field of characteristic zero. We consider the question which subsets of $M_n(F)$ can be images of noncommutative polynomials. We prove that a noncommutative polynomial $f$ has only finitely many similarity…

We study non-linear surjective mappings on subsets of ${\cal M}_n(F)$, which preserve the zeros of some fixed polynomials in noncommuting variables. Keywords: Matrix algebra, Multilinear polynomials, Preservers.

We find a basis for the $G$-graded identities of the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of characteristic $p>0$ with an elementary grading such that the neutral component corresponds to the diagonal of $M_n(K)$.

We show that for every subset $E$ of positive density in the set of integer square-matrices with zero traces, there exists an integer $k \geq 1$ such that the set of characteristic polynomials of matrices in $E-E$ contains the set of…

The main goal of this paper is to extend [J. Algebra Appl. 20 (2021), 2150074] to generalized quaternion algebras, even when these algebras are not necessarily division rings. More precisely, in such cases, the image of a multilinear…

Let K be a field and let M_n(K) denote the space of n x n matrices with entries in K. Let M be a subspace of M_n(K) of dimension d with the property that there are elements in M with non-zero determinant. Given a basis of M, we define the…