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

Abstract

Let p be a polynomial in non-commutative variables x1,x2,…,xn 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 Tm(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 Tt(K) but not of Tt+1(K), then p(Tm(K))⊆Tm(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 Tm(K)(t−1) can be written as sum of d many elements of p(Tm(K)). It has also been shown that the image of Tm(K)× under a word map is Zariski dense in Tm(K)×.