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

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.

## 1 Answer

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.

• Thanks. It was indeed similar to the proof of the nullstellensatz. Apr 6, 2020 at 14:36
• Actually, this has a name. It is the so-called Rabinowitsch trick: en.wikipedia.org/wiki/… Apr 6, 2020 at 14:40