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

Abstract

In the present paper we shall obtain the following result: Let $n\ge 2$ and $m\ge 1$ be integers. Let $p(x_1,\dots ,x_m)$ be a polynomial with zero constant term in non-commutative variables over an algebraically closed field K. Suppose that $1\le r\le n-1$, 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.

KEYWORDS:

• Algebraically closed field
• Fagundes-Mello conjecture
• polynomial
• upper triangular matrix algebra
• Zariski topology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 15A54
• 16R10
• 16S50

Acknowledgments

The author expresses thinks to my supervisor Professor Wang for useful suggestions. The author is grateful to the referee for some useful comments.