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.