Proposition

$\mathbf{V}(y - x^2, z - x^3)$ is the twisted cubic in $\mathbb{R}^3$.

Show that $\mathbf{V}((y - x^2)^2 + (z - x^3)^2)$ is also the twisted cubic.
Show that any variety $\mathbf{V}(I) \subset \mathbb{R}^n, I \subset \mathbb{R}[x_1, \cdots, x_n]$, can be defined by a single equation.

Solution

1

A point $(x, y, z) \in \mathbb{R}^3$ vanishes on $\{ y - x^2, z - x^3 \}$ if and only if it vanishes on $\{ (y - x^2)^2 + (z - x^3)^2 \}$. Thus $\mathbf{V}((y - x^2)^2 + (z - x^3)^2)$ is also the twisted cubic.

2

By Theorem 4 (Hilbert Basis Theorem) (P.77, Ideals, Varieties and Algorithms), $I$ has a finite generating set $\{ g_1, \cdots, g_k \}$. Then $\mathbf{V}(I) = \mathbf{V}(g_1^2 + \cdots + g_k^2)$ for the same reason as 1.