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

Problem:

Solution:

Consider a point $\vec{x} \in \mathbf{V}(f_1, \cdots, f_s)$, we know $f_1(\vec{x}) = \cdots = f_s(\vec{x}) = 0$.

Because $\langle f_1, \cdots, f_s \rangle = \langle g_1, \cdots, g_t \rangle$, therefore $g_k = h_1f_1 + \cdots + h_sf_s$ for all $k \in [1, t]$.

So $g_k(\vec{x}) = h_1f_1(\vec{x}) + \cdots + h_sf_s(\vec{x}) = 0$, so the point $\vec{x} \in \mathbf{V}(g_1, \cdots, g_t)$.

We have just shown $\mathbf{V}(f_1, \cdots, f_s) \subset \mathbf{V}(g_1, \cdots, g_t)$.

By symmetry, we also know $\mathbf{V}(g_1, \cdots, g_t) \subset \mathbf{V}(f_1, \cdots, f_s)$, so the two varieties are equal.