The image of multilinear polynomials evaluated on $3\times 3$ upper triangular matrices

TL;DR

This paper characterizes the images of multilinear polynomials of any degree when evaluated on the algebra of 3x3 upper triangular matrices over an infinite field, providing a comprehensive understanding of their behavior.

Contribution

It offers a complete description of the images of multilinear polynomials on 3x3 upper triangular matrices, extending previous results to arbitrary degrees.

Findings

Describes the possible images of multilinear polynomials on 3x3 upper triangular matrices.

Provides a classification of these images over infinite fields.

Extends known results to polynomials of any degree.

Abstract

We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3\times 3$ upper triangular matrix algebra over an infinite field.