Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices

TL;DR

The paper proves that the image of any nonzero multilinear polynomial over an infinite field includes all trace zero matrices in finitary matrix algebras, revealing a broad algebraic property.

Contribution

It establishes that for any positive integer, the image of a multilinear polynomial on finitary matrices encompasses all trace zero matrices, a novel generalization in matrix algebra.

Findings

Image of f contains all trace zero matrices in finitary algebra.

For each dimension, there exists a size s where the image includes all trace zero matrices.

The result applies to all finitary matrices over infinite fields.

Abstract

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$, contains all trace zero $d \times d$ matrices. In particular, the image of $f$ in the algebra of all finitary matrices contains all trace zero finitary matrices.