The Waring Problem for Matrix Algebras, II

TL;DR

This paper extends the Waring problem to matrix algebras, showing that under certain conditions, any trace-zero matrix can be expressed as a linear combination of images of a noncommutative polynomial.

Contribution

It proves a new result on the Waring problem for matrix algebras, generalizing previous work by considering noncommutative polynomials and specific linear combinations.

Findings

Any trace-zero matrix can be expressed as a linear combination of polynomial images.

The result holds for matrices of size n ≥ m-1 with specific coefficients summing to zero.

The work applies to noncommutative polynomials over algebraically closed fields of characteristic zero.

Abstract

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $\alpha_1,\alpha_2,\alpha_3$ are nonzero elements from $F$ such that $\alpha_1+\alpha_2+\alpha_3=0$, then every trace zero $n\times n$ matrix over $F$ can be written as $\alpha_1 A_1+\alpha_2A_2+\alpha_3A_3$ for some $A_i$ in the image of $f$ in $M_n(F)$.