On the image of a noncommutative polynomial

Abstract

Let F be an algebraically closed field of characteristic zero. We consider the question which subsets of Mn(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 GLn(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 Lvovʼs conjecture for multilinear Lie polynomials of degree at most 4 affirmatively.