The images of multilinear polynomials evaluated on 3 × 3 matrices

Let p be a multilinear polynomial in several noncommuting variables, with coefficients in an algebraically closed field K of arbitrary characteristic. In this paper we classify the possible images of p evaluated on 3 × 3 matrices. The image is one of the following: • {0}, • the set of scalar matrices, • a (Zariski-) dense subset of sl3(K), the matrices of trace 0, • a dense subset of M₃(K), • the set of 3-scalar matrices (i.e., matrices having eigenvalues (β, βε, βε²) where ε is a cube root of 1), or • the set of scalars plus 3-scalar matrices.