# Can I define a graded ring in sage?

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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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.

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

( 2016-06-21 18:50:07 +0200 )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.

( 2016-06-22 08:47:59 +0200 )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.

( 2016-06-22 09:00:54 +0200 )edit