Polynomials of small degree evaluated on matrices

TL;DR

This paper generalizes Shoda's theorem by proving that all nonzero multilinear polynomials of degree up to 3 evaluated on matrices produce all trace-zero matrices, extending the class of polynomials with this property.

Contribution

The authors extend Shoda's theorem to include all nonzero multilinear polynomials of degree at most 3 evaluated on matrices.

Findings

All nonzero multilinear polynomials of degree ≤ 3 have the property.

The set of polynomial values includes all matrices with trace zero.

Conjecture for higher degrees m ≤ n+1.

Abstract

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.