Power-central polynomials on matrices

Abstract

Any multilinear non-central polynomial $p$ (in several noncommuting variables) takes on values of degree $n$ in the matrix algebra $M_n(F)$ over an infinite field $F$. The polynomial $p$ is called {\it $ν$-central} for $M_n(F)$ if $p^ν$ takes on only scalar values, with $k$ minimal such. Multilinear $ν$-central polynomials do not exist for any $ν$ with $n>3$, thereby answering a question of Drensky. Saltman proved that an arbitrary polynomial $p$ cannot be $ν$-central for $M_n(F)$ for $n$ odd unless $n$ is prime; we show for $n$ even, that $ν$ must be 2.