TY - JOUR

T1 - Power-central polynomials on matrices

AU - Kanel-Belov, Alexey

AU - Malev, Sergey

AU - Rowen, Louis

N1 - Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84957601969&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2015.11.001

DO - 10.1016/j.jpaa.2015.11.001

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AN - SCOPUS:84957601969

SN - 0022-4049

VL - 220

SP - 2164

EP - 2176

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 6

ER -