Images of multilinear polynomials on $n\times n$ upper triangular matrices over infinite fields

TL;DR

This paper proves that the image of multilinear polynomials on upper triangular matrices over infinite fields equals a power of the Jacobson ideal, confirming a conjecture and introducing the concept of commutator-degree.

Contribution

It establishes the image of multilinear polynomials on $UT_n(K)$ as powers of the Jacobson ideal and introduces the notion of commutator-degree for such polynomials.

Findings

The image of multilinear polynomials on $UT_n(K)$ is $J^r$.

Multilinear polynomials of commutator-degree $r$ have image $J^r$.

The paper confirms the Lvov-Kaplansky conjecture for $UT_n(K)$.

Abstract

In this paper we prove that the image of multilinear polynomials evaluated on the algebra $UT_n(K)$ of $n\times n$ upper triangular matrices over an infinite field $K$ equals $J^r$, a power of its Jacobson ideal $J=J(UT_n(K))$. In particular, this shows that the analogue of the Lvov-Kaplansky conjecture for $UT_n(K)$ is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree $r$ in terms of its coefficients. It turns out that the image of a multilinear polynomial $f$ on $UT_n(K)$ is $J^r$ if and only if $f$ has commutator degree $r$.