# 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

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

add a comment

1

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?
```

1

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]
```

Asked: **
2013-06-20 08:03:04 -0600
**

Seen: **1,970 times**

Last updated: **Jun 23 '13**

Multivariate Polynomials over Rational Function Fields

Basis of invariant polynomial system

Checking the progress of a calculation

Problem computing Grobner basis

Compute Groebner Basis of an ideal that includes parameters

System of polynomial inequalities

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.