The images of multilinear polynomials evaluated on 3 × 3 matrices

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2016-01, Vol.144 (1), p.7-19
Main Authors: Kanel-Belov, Alexey, Malev, Sergey, Rowen, Louis
Format: Article
Language:English
Subjects:
A. ALGEBRA, NUMBER THEORY, AND COMBINATORICS
Online Access:Get full text
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Summary: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 M 3(K), the set of 3-scalar matrices (i.e., matrices having eigenvalues (β, βε, βε 2) where ε is a cube root of 1), or the set of scalars plus 3-scalar matrices. 2010 Mathematics Subject Classification. Primary 16R99, 15A24, 17B60; Secondary 16R30. Key words and phrases. Noncommutative polynomial, image, multilinear, matrices.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/12478