Agradecimentos: Thanks are due to the Referee whose careful reading and suggestions, accepted with gratitude, helped us improve the exposition of the results in our paper. P. Fagundes was supported by São Paulo Research Foundation (FAPESP), Grant No. 2019/16994-1. P. Koshlukov was partially supported by São Paulo Research Foundation (FAPESP), Grant No. 2018/23690-6 and by CNPq Grant No. 302238/2019-0 Ver menos

Abstract: In this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices UTn. For positive integers q <= n, we classify these images on UTn endowed with a particular elementary Z(q)-grading. As a consequence, we obtain the images of multilinear graded polynomials on UTn with the natural Z(n)-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras UT2 and UT3, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra UJ(2), and also for UJ(3) endowed with the natural elementary Z(3)-grading Ver menos