# On the image of a noncommutative polynomial

- Authors
- Type
- Preprint
- Publication Date
- Jan 16, 2013
- Submission Date
- Dec 19, 2012
- Identifiers
- arXiv ID: 1212.4600
- Source
- arXiv
- License
- Yellow
- External links

## Abstract

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.