The Waring Problem for Matrix Algebras

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".