Prove that there is a finite subset of these polynomials whose zeros define the same locus.

Question

I am attempting to solve Ch 14 Problem 6.1 from Artin's Algebra textbook.

Let $V\subset\mathbb{C}^n$ be the locus of common zeros of an infinite set of polynomials $f_1, f_2, f_3, \cdots$

Prove that there is a finite subset of these polynomials whose zeros define the same locus.

Now, the problem is from a section about Noetherian Rings, so I assume that this will be key to the solution. Let $I=(f_1,f_2,...).$ By the Hilbert Basis Theorem, I see $\mathbb{C}[x_1,...,x_n]$ is a Noetherian ring, so $I$ is finitely generated, WLOG say $I=(f_1,f_2,...,f_m)$. But I am unsure of what to do next. Please help.

Answer 1

Let $S=\{f_1,f_2,...\}$ and let $K=\{f_1,...,f_m\}$. Given any subset $T$ of $\mathbb{C}[x_1,...,x_n]$, define $f(T)=\{y\in\mathbb{C}^n:\forall t\in T[t(y)=0]\}$.

Now, you just need to show $f(S)=f(I)$ and $f(I)=f(K)$.

Answer 2

You are done! $V(I)=V(f_1,\dots,f_n)$, and so the vanishing locus depends only on finitely many polynomials.

In particular, you should show that $V=V(I)=V(f_1,\dots,f_n)=V(f_1) \cap V (f_2)\dots \cap V(f_n)$.

The final statement should be read as "every vanishing locus is the intersection of finitely many hypersurfaces," which is of course equivalent to your question.