TY - JOUR

T1 - The images of multilinear polynomials evaluated on 3 × 3 matrices

AU - Kanel-Belov, Alexey

AU - Malev, Sergey

AU - Rowen, Louis

N1 - Publisher Copyright:
© 2015 American Mathematical Society.

PY - 2016/1

Y1 - 2016/1

N2 - Let p be a multilinear polynomial in several noncommuting variables, with coefficients in an algebraically closed field K of arbitrary characteristic. In this paper we classify the possible images of p evaluated on 3 × 3 matrices. The image is one of the following: • {0}, • the set of scalar matrices, • a (Zariski-) dense subset of sl3(K), the matrices of trace 0, • a dense subset of M3(K), • the set of 3-scalar matrices (i.e., matrices having eigenvalues (β, βε, βε2) where ε is a cube root of 1), or • the set of scalars plus 3-scalar matrices.

AB - Let p be a multilinear polynomial in several noncommuting variables, with coefficients in an algebraically closed field K of arbitrary characteristic. In this paper we classify the possible images of p evaluated on 3 × 3 matrices. The image is one of the following: • {0}, • the set of scalar matrices, • a (Zariski-) dense subset of sl3(K), the matrices of trace 0, • a dense subset of M3(K), • the set of 3-scalar matrices (i.e., matrices having eigenvalues (β, βε, βε2) where ε is a cube root of 1), or • the set of scalars plus 3-scalar matrices.

KW - Image

KW - Matrices

KW - Multilinear

KW - Noncommutative polynomial

UR - http://www.scopus.com/inward/record.url?scp=84945294113&partnerID=8YFLogxK

U2 - 10.1090/proc/12478

DO - 10.1090/proc/12478

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AN - SCOPUS:84945294113

SN - 0002-9939

VL - 144

SP - 7

EP - 19

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -