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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.<x,y,z> = 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)
<class 'sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal'>
sage: I.parent()
Monoid of ideals of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: type(B)
<class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>
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?