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

Abstract

In the present paper we shall obtain the following result: Let n ≥ 2 and m ≥ 1 be integers. Let p ( x 1 , … , x m ) 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 ( T n ( K ) ) is a dense subset of T n ( 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.