Polynomials of small degree evaluated on matrices

A celebrated theorem of Shoda states that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or, equivalently, that the set of values of the polynomial f(x,y)=xy-yx on the nxn-matrix K-algebra contains all matrices with trace 0. We generalize Shoda's theorem by showing that every nonzero multilinear polynomial of degree at most 3, with coefficients in K, has this property. We further conjecture that this holds for every nonzero multilinear polynomial with coefficients in K of degree m, provided that m is at most n+1.

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