UTM Ideals Varieties and Algorithm - Chapter 1 Section 4 Exercise 5

Problem:

Solution:

This is easy.

In exercise 3, we just shown $\langle x + xy, y + xy, x^2, y^2 \rangle = \langle x, y \rangle$.
In exercise 4, we just shown if $\langle f_1, \cdots, f_s \rangle = \langle g_1, \cdots, g_t \rangle$, then $\mathbf{V}(f_1, \cdots, f_s) = \mathbf{V}(g_1, \cdots, g_t)$.

This is just applying the two facts we have just proved.