require "algebra"
P = MPolynomial(Rational, "xyz")
x, y, z = P.vars("xyz")
f1 = x**2 + y**2 + z**2 -1
f2 = x**2 + z**2 - y
f3 = x - z
p Groebner.basis([f1, f2, f3])
#=> [x - z, y - 2z^2, z^4 + 1/2z^2 - 1/4]