Images of graded polynomials on matrix algebras

The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra over a field K endowed with its canonical -grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on matrices, where n is a prime number. We confirm such conjecture for polynomials of degree 2 over when K is a quadratically closed field of characteristic zero or greater than n and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on .