The images of non-commutative polynomials evaluated on 2 × 2 matrices

Alexey Kanel-Belov, Sergey Malev, Louis Rowen

Research output: Contribution to journalArticlepeer-review

58 Scopus citations


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 Mn(K) of n by n matrices is either zero, or the set of scalar matrices, or the set sln(K) of matrices of trace 0, or all of Mn(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 Mn(K).

Original languageEnglish
Pages (from-to)465-478
Number of pages14
JournalProceedings of the American Mathematical Society
Issue number2
StatePublished - 2012


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