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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.

  • References
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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