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 Mn(F) over an infinite field F. The polynomial p is called ν-central for Mn(F) if pν takes on only scalar values, with ν minimal such. Multilinear ν-central polynomials do not exist for any ν, with n>3, answering a question of Drensky and Spenko.Saltman proved a result implying that a non-central polynomial p cannot be ν-central for Mn(F), for n odd, unless ν is a product of distinct odd primes and n=mν with m prime to ν; we extend this by showing for n even, that ν cannot be divisible by 4.