Proposition

Let $V = V(y - x)$ in $\mathbb{R}^2$ and let $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the polynomial mapping represented by $\phi(x, y) = (x^2 - y, y^2, x - 3y^2)$. The image of $V$ under $\phi$ is a variety in $\mathbb{R}^3$. Find a system of equations defining the image of $\phi$.

Solution

$V = \{ (a, a) \mid a \in \mathbb{R} \}$. Thus $\phi(V) = \{ (a^2 - a, a^2, a - 3a^2) \mid a \in \mathbb{R} \}$. Then we can regard $G(a) = (a^2 - a, a^2, a - 3a^2)$ to be a polynomial mapping from $\mathbb{R}$ into $\mathbb{R}^3$. Then $\phi(V) = G(\mathbb{R})$.

from sympy import *
from sympy.polys.orderings import monomial_key

x, y, z, a = symbols('x y z a')

print(groebner([x - a * a + a, y - a * a, z - a + 3 * a * a], a, z, y, x, order='lex'))

gives GroebnerBasis([a + x - y, x + 2*y + z, x**2 - 2*x*y + y**2 - y], a, z, y, x, domain='ZZ', order='lex').

By Theorem 1 (Polynomial Implicitization) on P.134 (Ideals, Varieties, and Algorithms), $V(x + 2y + z, x^2 - 2xy + y^2 - y)$ is the smallest variety in $\mathbb{R}^3$ containing $\phi(V)$. Since it is given that $\phi(V)$ is a variety, $\phi(V) = V(x + 2y + z, x^2 - 2xy + y^2 - y)$.