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

Abstract

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 × 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 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.