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.Fri, 18 Jun 2021 10:54:18 +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.EstanislaoFri, 18 Jun 2021 10:54:18 +0200https://ask.sagemath.org/question/57617/does sage allow computation of a groebner basis of an ideal J in the quotient ring Z/pZ[X_1,...X_r]/I?https://ask.sagemath.org/question/52493/does-sage-allow-computation-of-a-groebner-basis-of-an-ideal-j-in-the-quotient-ring-zpzx_1x_ri/I define the following rings and ideals:
sage: F = ZZ.quo(3*ZZ); F
Ring of integers modulo 3
sage: A.<X, Y, Z> = PolynomialRing(F); A
Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
sage: I = ideal(X^2 - 1, Y^2 - 1, Z^2 - 1); I
Ideal (X^2 + 2, Y^2 + 2, Z^2 + 2)
of Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
sage: R = A.quotient_ring(I); R
Quotient of Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
by the ideal (X^2 + 2, Y^2 + 2, Z^2 + 2)
sage: x, y, z = R.gens()
sage: J = ideal(x*y + z, x + x*z, x*z + y); J
Ideal (Xbar*Ybar + Zbar, Xbar*Zbar + Xbar, Xbar*Zbar + Ybar)
of Quotient of Multivariate Polynomial Ring in X, Y, Z
over Ring of integers modulo 3
by the ideal (X^2 + 2, Y^2 + 2, Z^2 + 2)
I then compute:
sage: B = J.groebner_basis()
and examine the result of that computation:
sage: B
[Xbar + 2*Ybar, Zbar + 1]
Is it really the Groebner basis of `J`?
Did Sage really compute in A/I and not in A?andriamThu, 16 Jul 2020 11:50:36 +0200https://ask.sagemath.org/question/52493/How do I deal with large, hex numbers in Sage?https://ask.sagemath.org/question/26061/how-do-i-deal-with-large-hex-numbers-in-sage/ How do I deal with large, hex numbers in Sage (say ones with 256 or 512 bytes)? How would I import them from a CSV file?
(cf. [this related answer](http://ask.sagemath.org/question/24708/how-to-enter-very-large-numbers/?answer=24710#post-id-24710))GeremiaSat, 07 Mar 2015 17:47:47 +0100https://ask.sagemath.org/question/26061/