The image of polynomials and Waring type problems on upper triangular matrix algebras

TL;DR

This paper investigates the images of non-commutative polynomials over upper triangular matrix algebras, establishing conditions for their containment, density, and representation as sums, thus advancing understanding of polynomial identities and Waring type problems in matrix algebras.

Contribution

It introduces multi-index p-inductive polynomials and characterizes the images of polynomials that are identities of certain upper triangular matrix algebras, providing new insights into their structure and density.

Findings

If p is an identity of T_t(K) but not T_{t+1}(K), then p(T_m(K)) is contained in T_m(K)^{(t-1)}.

The image of T_m(K)^ imes under a word map is Zariski dense in T_m(K)^ imes.

Existence of d such that every element of T_m(K)^{(t-1)} can be written as a sum of d elements from p(T_m(K)).

Abstract

Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices $T_m(K)$. We introduce a family of polynomials called multi-index $p$-inductive polynomials for a given polynomial $p$. Using this family we will show that, if $p$ is a polynomial identity of $T_t(K)$ but not of $T_{t+1}(K)$, then $p \left(T_m(K)\right)\subseteq T_m(K)^{(t-1)}$. Equality is achieved in the case $t=1,~m-1$ and an example has been provided to show that equality does not hold in general. We further prove existence of $d$ such that each element of $T_m(K)^{(t-1)}$ can be written as sum of $d$ many elements of $p\left( T_m(K) \right)$. It has also been shown that the image of $T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$.