ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 19 Jun 2021 09:11:58 +0200Is it possible to compute a Gröbner basis of an ideal of a graded commutative algebra in SageMathhttps://ask.sagemath.org/question/57617/is-it-possible-to-compute-a-grobner-basis-of-an-ideal-of-a-graded-commutative-algebra-in-sagemath/Let me start by saying that I am a newbie to Sage.
Let us say I have a graded commutative algebra `A` using the command
`GradedCommutativeAlgebra`, and an ideal `I` of `A`.
For instance, something like the following (but this is just a toy example!):
sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=((1,1), (2,1), (3,2))
sage: I = A.ideal([z*y - x*y*y])
I would like to get a Gröbner basis of `I` from SageMath
(not for the previous example, which is immediate).
I know how to do this for polynomial algebras, but for graded
commutative algebras constructed using `GradedCommutativeAlgebra`
this does not seem to work. Is it possible?
Thanks in advance!
EDIT: I slightly changed the previous example to avoid any misunderstanding. I remark that, if one forgets the multidegree of the algebras, the algebras I am interested in (and produced by the command `GradedCommutativeAlgebra`) are super commutative for the underlying Z/2Z grading. In particular, in the previous example, we have `z*z = 0` in `A`, because `z` has total odd degree, and `z*y = - y * z` in `A`, since `y` also has odd degree.Fri, 18 Jun 2021 10:54:18 +0200https://ask.sagemath.org/question/57617/is-it-possible-to-compute-a-grobner-basis-of-an-ideal-of-a-graded-commutative-algebra-in-sagemath/Comment by FrédéricC for <p>Let me start by saying that I am a newbie to Sage. </p>
<p>Let us say I have a graded commutative algebra <code>A</code> using the command
<code>GradedCommutativeAlgebra</code>, and an ideal <code>I</code> of <code>A</code>.</p>
<p>For instance, something like the following (but this is just a toy example!):</p>
<pre><code>sage: A.<x,y,z> = GradedCommutativeAlgebra(QQ, degrees=((1,1), (2,1), (3,2))
sage: I = A.ideal([z*y - x*y*y])
</code></pre>
<p>I would like to get a Gröbner basis of <code>I</code> from SageMath
(not for the previous example, which is immediate).</p>
<p>I know how to do this for polynomial algebras, but for graded
commutative algebras constructed using <code>GradedCommutativeAlgebra</code>
this does not seem to work. Is it possible? </p>
<p>Thanks in advance!</p>
<p>EDIT: I slightly changed the previous example to avoid any misunderstanding. I remark that, if one forgets the multidegree of the algebras, the algebras I am interested in (and produced by the command <code>GradedCommutativeAlgebra</code>) are super commutative for the underlying Z/2Z grading. In particular, in the previous example, we have <code>z*z = 0</code> in <code>A</code>, because <code>z</code> has total odd degree, and <code>z*y = - y * z</code> in <code>A</code>, since <code>y</code> also has odd degree.</p>
https://ask.sagemath.org/question/57617/is-it-possible-to-compute-a-grobner-basis-of-an-ideal-of-a-graded-commutative-algebra-in-sagemath/?comment=57622#post-id-57622Do you really mean to use these super-commutative algebras ? or rather a standard grading, where commutation between variables does not involve their degrees ?Sat, 19 Jun 2021 09:11:58 +0200https://ask.sagemath.org/question/57617/is-it-possible-to-compute-a-grobner-basis-of-an-ideal-of-a-graded-commutative-algebra-in-sagemath/?comment=57622#post-id-57622