Proposition

Determine whether the following polynomials lie in the following radicals. If the answer is yes, what is the smallest power of the polynomial that lies in the ideal?

Is $x + y \in \sqrt{\ev{x^3, y^3, xy(x + y)}}$?
Is $x^2 + 3xz \in \sqrt{\ev{x + z, x^2y, x - z^2}}$?

Solution

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

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

print(groebner([x**3, y**3, x*y*(x + y), 1 - a * (x + y)], x, y, z, a, order='lex'))
print(groebner([x + z, x**2*y, x - z**2, 1 - a * (x**2 + 3*x*z)], x, y, z, a, order='lex'))

gives

GroebnerBasis([1], x, y, z, a, domain='ZZ', order='lex')
GroebnerBasis([x - 1, y, z + 1, 2*a + 1], x, y, z, a, domain='ZZ', order='lex')

, so the answer to 1 is yes and 2 is no by the radical membership algorithm described on P.185 of Ideals, Varieties and Algorithms.

$x^3 + y^3 + 3xy(x + y) = (x + y)^3$. Moreover, each monomial of $x^3$, $y^3$ and $xy(x + y)$ has a total degree of 3, so the smallest power of $x + y$ that lies in the ideal cannot be less than 3. Therefore, the answer is 3.