On Fagundes-Mello conjecture

Abstract

We define the β-index of a nonzero multilinear polynomial, which gives a kind of classification of a nonzero multilinear polynomial. By using the β-index we give a complete description of the images of multilinear polynomials in non-commutative variables evaluated on upper triangular matrix algebras, which solves the Fagundes-Mello Conjecture proposed by Fagundes and Mello in 2019. The Fagundes-Mello Conjecture is an important variation of the old and famous Lvov-Kaplansky Conjecture.