On Multilinear Polynomials In Four Variables Evaluated On Matrices

TL;DR

This paper proves that any multilinear polynomial in four variables evaluated on matrices over an algebraically closed field of characteristic zero produces all trace-zero matrices, revealing a broad image property.

Contribution

It establishes that the image of any such polynomial includes all trace-zero matrices, extending understanding of polynomial evaluations on matrix rings.

Findings

The image of multilinear polynomials in four variables on matrices contains all trace-zero matrices.

This result holds over algebraically closed fields of characteristic zero.

It advances the theory of polynomial evaluations in matrix algebra.

Abstract

Let $K$ be an algebraically closed field of characteristic $0$ and let $M_n(K)$, $n \ge 3$, be the matrix ring over $K$. We will show that the image of any multilinear polynomial in four variables evaluated on $M_n(K)$ contains all matrices of trace $0$.