$g^l\in \langle f_1,\dots,f_k \rangle$, if the ideal generated by $f_1,\dots, f_k, gy -1$ in $\Bbb C[x_1,\dots,x_n, y]$ contains $1$

Question

Let $f_1,\dots , f_k$ be polynomials in $\Bbb C[x_1,\dots,x_n]$. I want to show that for a polynomial $g\in \Bbb C[x_1,\dots,x_n]$, $g^l$ is contained in the ideal generated by $f_1,\dots, f_k$ for some $l$ if the ideal generated by $f_1,\dots, f_k, gy -1$ in $\Bbb C[x_1,\dots,x_n, y]$ contains $1$.

Because of the term $gy-1$, I thought this may be similar to the proof of Hilbert Nullstellensatz, so I tried to mimick the proof, but I can't see nothing. Any hints? Thanks in advance.

Answer 1

Write $h_1f_1+\cdots +h_kf_k + h(gy-1)=1$ where the $h_i$'s and $h$ are in $\mathbf{C}[x_1,\dots,x_n,y]$. Now, "replace" $y$ by $g^{-1}$ and clear the denominators of the subsequent left hand side in a clever way: you will get your result.