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

TL;DR

This paper proves Kaplansky's conjecture for the image of non-commutative polynomials evaluated on 2x2 matrices over the real numbers, extending understanding of polynomial images in matrix algebras.

Contribution

The paper verifies Kaplansky's conjecture for 2x2 matrices over the real numbers and semi-homogeneous polynomials over arbitrary fields, advancing the classification of polynomial images.

Findings

Kaplansky's conjecture holds for 2x2 matrices over ℝ.

The image of semi-homogeneous polynomials is classified.

Partial results are obtained for arbitrary fields.

Abstract

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 $K$.