Images of linear polynomials on upper triangular matrix algebras

The Fagundes-Mello conjecture asserts that every multilinear polynomial on upper triangular matrix algebras is a vector space, which is an improtant variation of the old and famous Lvov-Kaplansky conjecture. The goal of the paper is to give a description of the images of linear polynomials with zero constant term on the upper triangular matrix algebra under a mild condition on the ground field. As a consequence we improve all results on the Fagundes-Mello conjecture. As another consequence we improve some results by Fagundes and Koshlukov on the images of multilinear graded polynomials on upper triangular matrix algebras.