About a corollary of the Briançon-Skoda theorem

Question

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.

Answer 1

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. 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, 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!