What is the condition that k complex polynomials in k variables have a finite number of common zeros?
Background: The following is a version of Bezout's theorem:
Let there be 2 complex polynomials in 2 variables. If they are coprime, then the number of common zeros is finite and bounded by the product of their degrees.
I am interested in generalizing this to k complex polynomials in k variables. I already found a proof that if the number of common zeros is finite then it has the bound I want, for example, in this blog post by Terry Tao (Theorem 4). However, the condition for finiteness is not specified.
I thought the generalization is "pairwise coprime", but this is wrong. Let $k=3$ and consider the counterexample $x,y,x+y$ (as polynomials in $x,y,z$). As a less trivial counterexample, consider $x-z, y-z, x^2+y^2-2z^2$, for which any $x=y=z$ works.
The condition is that the polynomials generate an ideal of height $k$.
This is much harder to check for $k>2$ than $k=2$, because height $1$ primes are principal in $\mathbb{C}[X,Y]$, so we just need to check that our polynomials have no common factors.