On Panja-Prasad conjecture

In the present paper we shall investigate the Waring’s problem for upper triangular matrix algebras. The main result is the following: Let $n\ge 2$ and $m\ge 1$ be integers. Let $p(x_1,\dots ,x_m)$ be a noncommutative polynomial with zero constant term over an infinite field $K$. Let $T_n(K)$ be the set of all $n\times n$ upper triangular matrices over $K$. Suppose $1<r<n-1$, where $r$ is the order of $p$. We have that $p(T_n(K))+p(T_n(K))=J^r$, where $J$ is the Jacobson radical of $T_n(K)$. If $r=n-2$, then $p(T_n(K))=J^{n-2}$. This gives a definitive solution of a conjecture proposed by Panja and Prasad.