TY - JOUR

T1 - The images of non-commutative polynomials evaluated on 2 × 2 matrices

AU - Kanel-Belov, Alexey

AU - Malev, Sergey

AU - Rowen, Louis

PY - 2012

Y1 - 2012

N2 - Let p be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field K of any characteristic. It has been conjectured that for any n, the image of p evaluated on the set Mn(K) of n by 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). We prove the conjecture for n = 2, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of Mn(K).

AB - Let p be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field K of any characteristic. It has been conjectured that for any n, the image of p evaluated on the set Mn(K) of n by 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). We prove the conjecture for n = 2, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of Mn(K).

UR - http://www.scopus.com/inward/record.url?scp=82255186709&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2011-10963-8

DO - 10.1090/S0002-9939-2011-10963-8

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AN - SCOPUS:82255186709

SN - 0002-9939

VL - 140

SP - 465

EP - 478

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 2

ER -