The following is a corollary of the Briançon-Skoda theorem:

If $R$ is a regular Noetherian ring of Krull dimension $d$ and $f_1,f_2,...,f_{d+1}\in R$. Then, $f_1^df_2^d...f_{d+1}^d \in (f_1^{d+1},f_2^{d+1},...,f_{d+1}^{d+1})R$

Now, given $R=k[[x_1,...,x_d]]$ where $k$ is a field, then $R$ is a regular local ring, so the corollary applies. I was wondering if there is direct proof of the above corollary for the case of the power series ring. I remember reading that there is a direct proof for the case when $f_1,...,f_{d+1}$ are polynomials. Can anyone point to a reference for this.