## The images of multilinear polynomials evaluated on $3\times 3$ matrices

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- by Alexey Kanel-Belov, Sergey Malev and Louis Rowen PDF
- Proc. Amer. Math. Soc.
**144**(2016), 7-19 Request permission

## 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 $3\times 3$ matrices. The image is one of the following:

{0},

the set of scalar matrices,

a (Zariski-)dense subset of $\operatorname {sl}_3(K)$, the matrices of trace 0,

a dense subset of $M_3(K)$,

the set of $3$-scalar matrices (i.e., matrices having eigenvalues $( \beta , \beta \varepsilon , \beta \varepsilon ^2)$ where $\varepsilon$ is a cube root of 1), or

the set of scalars plus $3$-scalar matrices.

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## Additional Information

**Alexey Kanel-Belov**- Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
- MR Author ID: 251623
- ORCID: 0000-0002-1371-7479
- Email: beloval@math.biu.ac.il
**Sergey Malev**- Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
- Email: malevs@math.biu.ac.il
**Louis Rowen**- Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
- MR Author ID: 151270
- Email: rowen@math.biu.ac.il
- Received by editor(s): June 30, 2013
- Received by editor(s) in revised form: December 29, 2013
- Published electronically: September 11, 2015
- Additional Notes: This work was supported by the Israel Science Foundation (grant no. 1207/12)

The second named author was supported by an Israeli Ministry of Immigrant Absorbtion scholarship. - Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 7-19 - MSC (2010): Primary 16R99, 15A24, 17B60; Secondary 16R30
- DOI: https://doi.org/10.1090/proc/12478
- MathSciNet review: 3415572