The Waring Problem for Matrix Algebras

TL;DR

This paper investigates the Waring problem for matrix algebras, showing that trace zero matrices can be expressed as sums of two matrices derived from a noncommutative polynomial, with the minimal number of summands being two.

Contribution

It establishes the minimal number of summands needed to represent trace zero matrices as sums of polynomial images in matrix algebras, extending understanding of polynomial identities.

Findings

Trace zero matrices are sums of two polynomial images.

One summand is insufficient for such representations.

The result applies to noncommutative polynomials that are neither identities nor central.

Abstract

If a noncommutative polynomial $f$ is neither an identity nor a central polynomial of $\mathcal A=M_n(\C)$, then every trace zero matrix in $\mathcal A$ can be written as a sum of two matrices from $f(\mathcal A)-f(\mathcal A)$. Moreover, "two" cannot be replaced by "one".