A new approach to the Lvov-Kaplansky conjecture through gradings

Abstract

In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture.

arXiv comment10 pages; comments are welcome

References in corpus (9)

• The images of non-commutative polynomials evaluated on $2\times 2$ matrices
• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• Power-central polynomials on matrices
• Polynomials of small degree evaluated on matrices
• On Multilinear Polynomials In Four Variables Evaluated On Matrices
• Images of multilinear polynomials on $n\times n$ upper triangular matrices over infinite fields
• The images of multilinear and semihomogeneous polynomials on the algebra of octonions
• The Mesyan Conjecture: a restatement and a correction
• Images of graded polynomials on matrix algebras