  In this article we show that if ${\cal V}$ is the variety of polynilpotent
groups of class row $(c_1,c_2,...,c_s),\ {\mathcal N}_{c_1,c_2,...,c_s}$, and
$G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n}{*}{\bf
{Z}}_{p^{\alpha_2}}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{\alpha_t} }$
is the $n$th nilpotent product of some cyclic $p$-groups, where $c_1\geq n$,
$\alpha_1 \geq \alpha_2 \geq...\geq \alpha_t $ and $ (q,p)=1$ for all primes
$q$ less than or equal to $n$, then $|{\mathcal
N}_{c_1,c_2,...,c_s}M(G)|=p^{d_m}$ if and only if $G\cong{\bf
{Z}}_{p}\stackrel{n}{*}{\bf
{Z}}_{p}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p }$ ($m$-copies), where
$m=\sum _{i=1}^t \alpha_i$ and $d_m=\chi_{c_s+1}(...(\chi_{c_2+1}(\sum_{j=1}^n
\chi_{c_1+j}(m)))...)$. Also, we extend the result to the multiple nilpotent
product $G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n_1}{*}{\bf
{Z}}_{p^{\alpha_2}}\stackrel{n_2}{*}...\stackrel{n_{t-1}}{*}{\bf{Z}}_{p^{\alpha_t}
}$, where $c_1\geq n_1\geq...\geq n_{t-1}$. Finally a similar result is given
for the $c$-nilpotent multiplier of $G\cong{\bf
{Z}}_{p^{\alpha_1}}\stackrel{n}{*}{\bf
{Z}}_{p^{\alpha_2}}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{\alpha_t}}$
with the different conditions $n \geq c$ and $ (q,p)=1$ for all primes $q$ less
than or equal to $n+c.$
