UTM Ideals Varieties and Algorithm - Chapter 1 Section 2 Exercise 8

Problem:

Solution:

Again, the hint is holding my hand. To be frank, here is what I thought without the hints

First, the variety must be written as a set of polynomials, so if I substitute $(0, 0)$ to the polynomials, then all the constant terms must be 0, then if I substitute $(1, 1)$ to the polynomials, then get least one of the polynomial has a non-zero coefficients. But then I haven't go any further than that.

With the hint, it is easy.

$g_k(x) = f_k(x, x)$ vanishes on infinitely many points, so it is the zero polynomial, so $f_k(1, 1) = g_k(1) = 0$, so we get the desired contradiction.

The hint goes too far. It could have just hint me to use the proposition ...