Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$ , the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$ . We prove the conjecture for $n=2$ , and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of $M_n(K)$ .

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 $3\times 3$ matrices. The image is one of the following: {0}, the set of scalar matrices, a (Zariski-)dense subset of $\operatorname {sl}_3(K)$ , the matrices of trace 0, a dense subset of $M_3(K)$ , the set of $3$ -scalar matrices (i.e., matrices having eigenvalues $( \beta , \beta \varepsilon , \beta \varepsilon ^2)$ where $\varepsilon$ is a cube root of 1), or the set of scalars plus $3$ -scalar matrices.

Kaplansky asked about the possible images of a polynomial f in several noncommuting variables. In this paper, we consider the case of f a Lie polynomial. We describe all the possible images of f in M₂(K) and provide an example of f whose image is the set of non-nilpotent trace zero matrices, together with 0. We provide an arithmetic criterion for this case. We also show that the standard polynomial s_k is not a Lie polynomial, for k>2.