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 M-n(F) over an infinite field F. The polynomial p is called v-central for M-n(F) if p(v) takes on only scalar values, with v minimal such. Multilinear v-central polynomials do not exist for any v, with n > 3, answering a question of Drensky and Spenko. Saltman proved a result implying that a non-central polynomial p cannot be v-central for M-n(F), for n odd, unless v is a product of distinct odd primes and n = mv with m prime to v; we extend this by showing for n even, that v cannot be divisible by 4. (C) 2015 Elsevier B.V. All rights reserved.