  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 $T_m(K)^\times$.
