The images of non-commutative polynomials evaluated on $2\times 2$ matrices

TL;DR

This paper proves a conjecture about the possible images of multilinear non-commutative polynomials evaluated on 2x2 matrices, showing they are limited to four specific sets.

Contribution

It confirms the conjecture for 2x2 matrices, characterizing the image of such polynomials over any quadratically closed field.

Findings

The image of multilinear polynomials on 2x2 matrices is either zero, scalar matrices, trace-zero matrices, or all matrices.

The conjecture is validated specifically for the case n=2.

The result holds over fields of any characteristic.

Abstract

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 $n=2$.