Alexey Kanel-Belov, Sergey Malev, Louis Rowen. The images of non-commutative polynomials evaluated on 2x2 matrices
Natural Sciences / Mathematics / Algebra
Submitted on: Aug 27, 2012, 22:39:35
Description: 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 $n=2$.