Polynomials of small degree evaluated on matrices

Abstract

A celebrated theorem of Shoda states that over any field (of characteristic ), every matrix with trace can be expressed as a commutator , or, equivalently, that the set of values of the polynomial on contains all matrices with trace . We generalize Shoda’s theorem by showing that every non-zero multilinear polynomial of degree at most , with coefficients in , has this property. We further conjecture that this holds for every non-zero multilinear polynomial with coefficients in of degree , provided that .