THE IMAGES OF MULTILINEAR POLYNOMIALS EVALUATED ON 3 x 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 x 3 matrices. The image is one of the following: {0}, the set of scalar matrices, a (Zariski-) dense subset of sl(3)(K), the matrices of trace 0, a dense subset of M-3(K), the set of 3-scalar matrices (i.e., matrices having eigenvalues (beta, beta epsilon, beta epsilon(2)) where epsilon is a cube root of 1), or the set of scalars plus 3-scalar matrices.