TY - JOUR

T1 - The images of non-commutative polynomials evaluated on 2 × 2 matrices over an arbitrary field

AU - Malev, Sergey

PY - 2014/9

Y1 - 2014/9

N2 - Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mn(K) of n × n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = reals and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

AB - Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mn(K) of n × n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = reals and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

KW - Kaplansky conjecture

KW - Non-commutative polynomials

KW - matrix algebras

KW - polynomial images

UR - http://www.scopus.com/inward/record.url?scp=85027941397&partnerID=8YFLogxK

U2 - 10.1142/S0219498814500054

DO - 10.1142/S0219498814500054

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AN - SCOPUS:85027941397

SN - 0219-4988

VL - 13

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

IS - 6

M1 - 1450004

ER -