On Panja-Prasad conjecture

TL;DR

This paper solves a conjecture related to Waring's problem for upper triangular matrix algebras, showing how polynomial images relate to powers of the Jacobson radical.

Contribution

It provides a definitive solution to the Panja-Prasad conjecture concerning polynomial images in upper triangular matrix algebras.

Findings

For r between 1 and n-1, p(T_n(K)) + p(T_n(K)) equals J^r.

When r equals n-2, p(T_n(K)) equals J^{n-2}.

The results clarify the structure of polynomial images in matrix algebras.

Abstract

In the present paper we shall investigate the Waring's problem for upper triangular matrix algebras. The main result is the following: Let $n\geq 2$ and $m\geq 1$ be integers. Let $p(x_1,\ldots,x_m)$ 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.