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

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.