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

By A. Kanel’-Belov, Sergey Malev, Louis H. Rowen. 2010

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Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been 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 $M_n(K)$. We prove the conjecture for $n=2$.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

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