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

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 this paper is the image kind algebra upper triangular matrices Tm(K). We introduce family polynomials called multi-index p-inductive for given p. Using we will show that, if identity Tt(K) but not Tt+1(K), then p(Tm(K))⊆Tm(K)(t−1). Equality achieved case t=1,m−1 and example has been provided to that equality does hold general. further prove existence d such each element Tm(K)(t−1) can written as sum many elements p(Tm(K)). It also shown Tm(K)× under word map Zariski dense Tm(K)×.