The images of multilinear polynomials evaluated on $3\times 3$ matrices
Alexei Kanel-Belov, Sergey Malev, Louis Rowen

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