2019-02-03 20:41:02 +0100 received badge ● Student (source) 2017-06-20 16:23:22 +0100 received badge ● Famous Question (source) 2016-11-29 12:10:30 +0100 received badge ● Notable Question (source) 2016-11-29 12:10:30 +0100 received badge ● Popular Question (source) 2016-04-10 19:54:20 +0100 received badge ● Notable Question (source) 2016-04-10 19:54:20 +0100 received badge ● Famous Question (source) 2016-04-10 19:54:20 +0100 received badge ● Popular Question (source) 2013-06-21 02:41:14 +0100 received badge ● Supporter (source) 2013-06-21 02:41:11 +0100 marked best answer Groebner basis The three polynomials used to describe the particular ideal in your example are a Groebner basis for it. sage: R. = PolynomialRing(QQ,3) sage: I = Ideal([x^2+y+z-1,x+y^2+z-1,x+y+z^2-1]) sage: B = I.groebner_basis(); B [x^2 + y + z - 1, y^2 + x + z - 1, z^2 + x + y - 1]  However, the ideal and its Groebner basis are not the same. Check their type and their parent. sage: type(I) sage: I.parent() Monoid of ideals of Multivariate Polynomial Ring in x, y, z over Rational Field sage: type(B) sage: B.parent() Category of sequences in Multivariate Polynomial Ring in x, y, z over Rational Field  Also, if you check the documentation for groebner_basis, you will find examples of families of polynomials who are not a Groebner basis for the ideal they generate. sage: I.groebner_basis?  2013-06-21 02:41:11 +0100 received badge ● Scholar (source) 2013-06-20 15:03:04 +0100 asked a question Groebner basis hello I'm trying to compute groebner basis for I=( x^2+y+z-1,x+y^2+z-1,x+y+z^2-1) in sage, but why the groebner basis of this ideal is same as ideal? thank you 2013-06-20 15:00:13 +0100 asked a question reduced groebner basis hello, how can I compute reduced groebner basis with out using buchberger algorithm in sage?