Abstract: Let $n\geq 2$ be a natural number, $1\leq p\leq \infty $ and $X$ a Banach space. We prove that if $X^{\ast }$ contains $\lambda $-uniformly copies of $ l_{p}^{k}$, then:
(i) $\mathcal{P}\left( ^{n}X\right) $ contains $c_{\mathbb{K} }\lambda ^{n}$-uniformly copies of $l_{\left( \frac{p^{\ast }}{n}\right)
^{\ast }}^{k}$ in the case $p^{\ast }>n$;
(ii) $\mathcal{P}\left( ^{n}X\right) $ contains\textit{\ }$\lambda ^{n}$ \textit{-}uniformly copies of $l_{\infty }^{k}$ in the case $p^{\ast }\leq n. $ This complete a result of S. Dineen's from 1995.

Key words: polynomials on Banach spaces, uniformly copies of $l_{p}^{n}$, finitely represented