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

Abstract

Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\dots ,x_n$ 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 $T_m(K{)}^{\times }$.

arXiv comment39 pages, 6 figures, and comments are welcome; Waring type problem is modified

References in corpus (6)

• The images of non-commutative polynomials evaluated on $2\times 2$ matrices
• The images of multilinear polynomials evaluated on $3\times 3$ matrices
• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• Power-central polynomials on matrices
• Images of multilinear polynomials of degree up to four on upper triangular matrices
• Images of multilinear polynomials on $n\times n$ upper triangular matrices over infinite fields

Cited by in corpus (1)

• Waring Problem For Triangular Matrix Algebra