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.



About a corollary of the Briançon-Skoda theorem
It depends on what do you mean by "direct". There is no elementary proof, as far as I known, even for polynomials when $d=2$, see page 8 of this [note](http://www.math.lsa.umich.edu/%7Ehochster/balt.ps). When $d=1$, the statement is an easy exercise: one can write $f\_1= da, f\_2=db$ with $(a,b) = R$ since $R$ is an UFD.


There are a few proofs of this very interesting theorem, analytic (the original one, over complex numbers), using duality theory ([Lipman-Sathaye](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-28/issue-2/Jacobian-ideals-and-a-theorem-of-Brianc-con-Skoda/10.1307/mmj/1029002510.full), for all regular rings) and reduction to characteristic $p$ + tight closure (Hochster-Huneke, when $R$ contains a field, see the note in the first paragraph for some exposition of this method) but I am not sure any of them can be called direct. Any new proof will be very exciting!


