Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three

Abstract

Given a positive integer d, the Kaplansky-Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f on the matrix algebra M_d(C) is a vector subspace. In this article the technique of using one-wiggle families of Sylvester's clock-and-shift matrices is championed to establish the conjecture for polynomials f of degree three when d is even or d<17.

v3: 17 pages (minor changes); supplementary material (a Mathematica notebook) is available from Other formats or can be found at http://www.math.auckland.ac.nz/~igorklep/publ.html

References in corpus (2)

• The images of Lie polynomials evaluated on matrices
• On Multilinear Polynomials In Four Variables Evaluated On Matrices

Cited by in corpus (5)

• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• The images of multilinear and semihomogeneous polynomials on the algebra of octonions
• Images of graded polynomials on matrix algebras
• On the image of polynomials evaluated on incidence algebras: a counter-example and a solution
• The L'vov-Kaplansky Conjecture for Polynomials of Degree Three