On the image of a noncommutative polynomial

TL;DR

This paper characterizes the images of noncommutative polynomials over algebraically closed fields, showing conditions for finiteness of similarity orbits, constructing specific polynomial images, and confirming Lvov's conjecture for certain multilinear Lie polynomials.

Contribution

It provides a complete characterization of polynomial images with finitely many similarity orbits and confirms Lvov's conjecture for multilinear Lie polynomials of degree up to 4.

Findings

Finitely many similarity orbits occur iff the polynomial is power-central.

Constructed polynomial images include the union of zero and a conjugation-invariant open set.

Confirmed Lvov's conjecture for multilinear Lie polynomials of degree ≤ 4.

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.