The images of non-commutative polynomials evaluated on 2×2 matrices

Let p p be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field K K of any characteristic. It has been conjectured that for any n n , the image of p p evaluated on the set M n ( K ) M_n(K) of n n by n n matrices is either zero, or the set of scalar matrices, or the set s l n ( K ) sl_n(K) of matrices of trace 0, or all of M n ( K ) M_n(K) . We prove the conjecture for n = 2 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 ) M_n(K) .