The images of non-commutative polynomials evaluated on $2\times 2$ matrices over an arbitrary field

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 $M_n(K)$. This conjecture was proved for $n=2$ when $K$ is closed under quadratic extensions. In this paper the conjecture is verified for $K=\mathbb{R}$ and $n=2$, also for semi-homogeneous polynomials $p$, with a partial solution for an arbitrary field $K$.

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