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Groebner basis

asked 2013-06-20 15:03:04 +0200

Roxana gravatar image

updated 2024-04-19 13:34:49 +0200

FrédéricC gravatar image


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

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answered 2013-06-20 21:41:06 +0200

slelievre gravatar image

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?
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answered 2013-06-23 21:00:35 +0200

Volker Braun gravatar image

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]


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]
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Asked: 2013-06-20 15:03:04 +0200

Seen: 4,277 times

Last updated: Jun 23 '13