  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 $n=2$.
