The images of non-commutative polynomials evaluated on 2 x 2 matrices over the real numbers

Wed, 13/11/2013 - 10:30
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 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). I prove the conjecture when K is the field of real numbers and
n=2, and give a partial solution for
an arbitrary field K.