Is it possible to compute a Gröbner basis of an ideal of a graded commutative algebra in SageMath

asked 2021-06-18 10:54:18 +0200

anonymous user


updated 2024-04-18 20:08:56 +0200

FrédéricC gravatar image

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.

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Do you really mean to use these super-commutative algebras ? or rather a standard grading, where commutation between variables does not involve their degrees ?

FrédéricC gravatar imageFrédéricC ( 2021-06-19 09:11:58 +0200 )edit