  In this paper we study the images of multilinear graded polynomials on the
graded algebra of upper triangular matrices UT_n. For positive integers q \leq
n, we classify these images on UT_n endowed with a particular elementary
Z_q-grading. As a consequence, we obtain the images of multilinear graded
polynomials on UT_n 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 UT_2 and UT_3, 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.
