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The images of non-commutative polynomials evaluated on $ 2\times 2$ matrices


Authors: Alexey Kanel-Belov, Sergey Malev and Louis Rowen
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
Published electronically: June 16, 2011
MathSciNet review: 2846315
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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
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
Email: rowen@math.biu.ac.il

DOI: https://doi.org/10.1090/S0002-9939-2011-10963-8
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
Article copyright: © Copyright 2011 American Mathematical Society