We describe the images of multilinear polynomials of degree up to four on the upper triangular matrix algebra.A famous open problem known as Lvov-Kaplansky's conjecture asserts: the image of a multilinear polynomial in noncommutative variables over a field K on the matrix algebra M n (K) is always a vector space [4].Recently, Kanel-Belov, Malev and Rowen [7] made a major breakthrough and solved the problem for n = 2.A special case on polynomials of degree two has been known for long time ([9] and [1]). Recently, Mesyan [8] and Buzinski and Winstanley [3] extended this result for nonzero multilinear polynomials of degree three and four, respectively.We will study the following variation of the Lvov-Kaplansky's conjecture:Conjecture 1 The image of a multilinear polynomial on the upper triangular matrix algebra is a vector space.In this paper, we will answer Conjecture 1 for polynomials of degree up to four. We point out that whereas in [3] and [8] the results describe conditions under which the image of a multilinear polynomial p, Im(p), contains a certain subset of M n (K), our results give the explicit forms of Im(p) on the upper triangular matrix algebra in each case.Throughout the paper UT n will denote the set of upper triangular matrices. The set of all strictly upper triangular matrices will be denoted by UT (0) n . More generally, if k ≥ 0, 1

The purpose of this paper is to describe the images of multilinear polynomials of arbitrary degree on the strictly upper triangular matrix algebra.

In this paper we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices U Tn. For positive integers q ≤ n, we classify these images on U Tn endowed with a particular elementary Zq-grading. As a consequence, we obtain the images of multilinear graded polynomials on U Tn with the natural Zn-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 U T 2 and U T 3 , for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra U J 2 , and also for U J 3 endowed with the natural elementary Z 3 -grading.

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