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
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