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Can I define a graded ring in sage?

asked 2016-06-20 14:40:28 -0500

done_with_fish gravatar image

I can define a custom grading on a polynomial ring in Macaulay2 with the command

S = QQ[x,y, Degrees => {{1},{2}}]

Can I define this ring in sage?

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answered 2016-06-21 01:38:33 -0500

B r u n o gravatar image

If I correctly understand the Macaulay2 command you provide, you can mimick the same behavior in SageMath as follows:

sage: T = TermOrder("wdeglex", (1,2))
sage: R = PolynomialRing(QQ, 'x,y', order=T)
sage: R
Multivariate Polynomial Ring in x, y over Rational Field
sage: x,y = R.gens()
sage: (x*y).degree()
3

You can find more informations on term orders in the documentation [1]. Several weighted term orders are available.

[1] http://doc.sagemath.org/html/en/refer...

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Ahh great! Also, is it possible to define a bidegree? TermOrder('wdeglex', ((1,0),(2,2))) returns a TypeError

done_with_fish gravatar imagedone_with_fish ( 2016-06-21 11:50:07 -0500 )edit

I do not think so. I urge you to propose a ticket on http://trac.sagemath.org to implement this kind of term orders.

B r u n o gravatar imageB r u n o ( 2016-06-22 01:47:59 -0500 )edit

Well, thinking a bit more about this, you can use matrix term orders:

sage: S = TermOrder('M(2,3,0,1)')
sage: S
Matrix term order with matrix
[2 3]
[0 1]

In such a way, the first variable will have weight (2,0) and the second (3,1). This order is used when you compare monomials. Unfortunately, the method degree only returns the degree with respect with the first weight of each variable rather than a tuple. I do not know how one can get the tuple.

B r u n o gravatar imageB r u n o ( 2016-06-22 02:00:54 -0500 )edit

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Asked: 2016-06-20 14:40:28 -0500

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Last updated: Jun 21 '16