A note on the image of polynomials on upper triangular matrix algebras

In the present paper we shall obtain the following result: Let n≥2 and m≥1 be integers. Let p(x1,…,xm) be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that 1≤r≤n−1, where r is the order of p. Then p(Tn(K)) is a dense subset of Tn(K)(r−1) (with respect to the Zariski topology). This is a solution of a problem presented by Gargate and de Mello and a supplement to a result obtained by Panja and Prasad in 2023.