On the image of a noncommutative polynomial

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 orbits modulo nonzero scalar multiplication in its image if and only if $f$ is power-central. The union of the zero matrix and a standard open set closed under conjugation by $GL_n(F)$ and nonzero scalar multiplication is shown to be the image of a noncommutative polynomial. We investigate the density of the images with respect to the Zariski topology. We also answer Lvov's conjecture for multilinear Lie polynomials of degree at most 4 affirmatively.Comment: 13 pages, accepted for publication in J. Algebr

Noncommutative polynomial maps

Accepté pour publication dans "Journal of Algebra and its applications"; 16 pages.Polynomial maps attached to polynomials of an Ore extension are naturally defi ned. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a fi nite fi eld F_q[t;S ], where S is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring F_q[x]

A Note on Values of Noncommutative Polynomials

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 an infinite dimensional Hilbert space.Comment: 4 page