Polynomials of small degree evaluated on matrices

摘要

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 M_n(K) contains all matrices with trace 0. We generalize Shoda's theorem by showing that every non-zero multilinear polynomial of degree at most 3, with coefficients in K, has this property.We further conjecture that this holds for every non-zero multilinear polynomial with coefficients in K of degreem, provided that m ? 1 ≤ n.