The images of multilinear polynomials evaluated on $3\times 3$ matrices

TL;DR

This paper classifies the possible images of multilinear polynomials evaluated on 3x3 matrices over algebraically closed fields, identifying six distinct types of image sets.

Contribution

It provides a complete classification of the images of multilinear polynomials on 3x3 matrices, extending understanding of polynomial evaluations in matrix algebra.

Findings

Six possible image types identified

Images include zero, scalar, trace-zero, dense in matrices, 3-scalar, and scalar plus 3-scalar matrices

Classification holds over algebraically closed fields of any characteristic

Abstract

Let $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in a algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\times 3$ matrices. The image is one of the following: \begin{itemize} \item \{0\}, \item the set of scalar matrices, \item a (Zariski) dense subset of $\sl_3(K)$, the matrices of trace 0, \item a dense subset of $M_3(K)$, \item the set of $3-$scalar matrices (i.e., matrices having eigenvalues $(\beta, \beta \varepsilon, \beta \varepsilon^2)$ where $\varepsilon$ is a cube root of 1), or \item the set of scalars plus $3-$scalar matrices.