Abstract
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.
Original language | English |
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Pages (from-to) | 7-19 |
Number of pages | 13 |
Journal | Proceedings of the American Mathematical Society |
Volume | 144 |
Issue number | 1 |
DOIs | |
State | Published - Jan 2016 |
Keywords
- Image
- Matrices
- Multilinear
- Noncommutative polynomial
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- General Mathematics