The images of multilinear polynomials evaluated on 3×3 matrices

Kanel-Belov, Alexey; Malev, Sergey; Rowen, Louis

doi:10.1090/proc/12478

The images of multilinear polynomials evaluated on $3\times 3$ matrices
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Proc. Amer. Math. Soc. 144 (2016), 7-19 Request permission

Abstract:

Let $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in an algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\times 3$ matrices. The image is one of the following:

{0},

the set of scalar matrices,

a (Zariski-)dense subset of $\operatorname {sl}_3(K)$, the matrices of trace 0,

a dense subset of $M_3(K)$,

the set of $3$-scalar matrices (i.e., matrices having eigenvalues $( \beta , \beta \varepsilon , \beta \varepsilon ^2)$ where $\varepsilon$ is a cube root of 1), or

the set of scalars plus $3$-scalar matrices.

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