Power-central polynomials on matrices

Any multilinear non-central polynomial p (in several noncommuting variables) takes on values of degree n   in the matrix algebra Mn(F)Mn(F) over an infinite field F. The polynomial p is called ν-central   for Mn(F)Mn(F) if pνpν takes on only scalar values, with ν minimal such. Multilinear ν-central polynomials do not exist for any ν  , with n>3n>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)Mn(F), for n odd, unless ν   is a product of distinct odd primes and n=mνn=mν with m prime to ν; we extend this by showing for n even, that ν cannot be divisible by 4.