A new approach to the Lvov-Kaplansky conjecture through gradings

TL;DR

This paper investigates the Lvov-Kaplansky conjecture by analyzing the images of noncommutative polynomials on graded matrix algebras, providing new conditions and an equivalent reformulation of the conjecture.

Contribution

It introduces a grading-based approach to study the conjecture, offering necessary and sufficient conditions for polynomial images and an equivalent statement to the original conjecture.

Findings

Characterization of central and trace zero polynomials via gradings

Equivalent reformulation of the Lvov-Kaplansky conjecture

Conditions for polynomial images on graded matrix algebras

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.