Polynomials of small degree evaluated on matrices

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.

9 pages

References in corpus (1)

• The images of non-commutative polynomials evaluated on $2\times 2$ matrices

Cited by in corpus (8)

• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• On Multilinear Polynomials In Four Variables Evaluated On Matrices
• Images of multilinear polynomials of degree up to four on upper triangular matrices
• Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three
• The Mesyan Conjecture: a restatement and a correction
• Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices
• A new approach to the Lvov-Kaplansky conjecture through gradings
• The L'vov-Kaplansky Conjecture for Polynomials of Degree Three