Proposition

Determine all solutions $(x, y) \in \mathbb{Q}^2$ of the system of equations

Also determine all solutions in $\mathbb{C}^2$.

Solution

Let $I = \ev{x^2 + 2y^2 - 2, x^2 + xy + y^2 - 2}$.

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

x, y = symbols('x y')

print(groebner([x**2 + 2*y**2 - 2, x**2 + x*y + y**2 - 2], x, y, order='lex'))

gives GroebnerBasis([x**2 + 2*y**2 - 2, x*y - y**2, 3*y**3 - 2*y], x, y, domain='ZZ', order='lex') By solving this from the last element in the basis, we get $(x, y) = (\pm \sqrt{2}, 0), (\sqrt{6}/3, \sqrt{6}/3), (-\sqrt{6}/3, -\sqrt{6}/3)$. Thus there are no solutions in $\mathbb{Q}^2$ and there are 3 solutions in $\mathbb{C}^2$.