The images of non-commutative polynomials evaluated on 2× 2 matrices over an arbitrary field

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Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mₙ(K) of n by n matrices is either zero, or the set of scalar matrices, or the set slₙ(K) of matrices of trace 0, or all of Mₙ(K). This conjecture was proved for n=2 when K is closed under quadratic extensions. In this paper the conjecture is verified for K=R and n=2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.