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Also note that the Groebner basis depends on the monomial ordering:
sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I = Ideal([x^2+y+z-1,x+y^2+z-1,x+y+z^2-1])
sage: I.groebner_basis()
[x + y + z^2 - 1, y^2 - y - z^2 + z, y*z^2 + 1/2*z^4 - 1/2*z^2, z^6 - 4*z^4 + 4*z^3 - z^2]
vs.
sage: S.<x,y,z> = PolynomialRing(QQ, 3, order='deglex')
sage: J = Ideal([x^2+y+z-1,x+y^2+z-1,x+y+z^2-1])
sage: J.groebner_basis()
[x^2 + y + z - 1, y^2 + x + z - 1, z^2 + x + y - 1]
