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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 non-commutative polynomials evaluated on $2\times 2$ matrices
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by Alexey Kanel-Belov, Sergey Malev and Louis Rowen PDF
Proc. Amer. Math. Soc. 140 (2012), 465-478 Request permission

Abstract:

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of $M_n(K)$.
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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: belova@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 1, 2010
  • Received by editor(s) in revised form: November 29, 2010
  • Published electronically: June 16, 2011
  • Additional Notes: This work was financially supported by the Israel Science Foundation (grant No. 1178/06). The authors are grateful to V. Kulyamin, V. Latyshev, A. Mihalev, E. Plotkin, and L. Small for useful comments. Latyshev and Mihalev indicated that the problem was originally posed by I. Kaplansky.
  • Communicated by: Birge Huisgen-Zimmermann
  • © Copyright 2011 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 465-478
  • MSC (2010): Primary 16R30, 16R99; Secondary 16S50
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10963-8
  • MathSciNet review: 2846315