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 $\nu$-central} for $M_n(F)$ if $p^\nu$ takes on only scalar values, with $k$ minimal such. Multilinear $\nu$-central polynomials do not exist for any $\nu$ with $n>3$, thereby answering a question of Drensky. Saltman proved that an arbitrary polynomial $p$ cannot be $\nu$-central for $M_n(F)$ for $n$ odd unless $n$ is prime; we show for $n$ even, that $\nu$ must be 2.