Can sage compute a basis for the graded parts of a graded ring?

PolynomialRing

asked 2016-08-25 05:17:12 +0100

done_with_fish
273 ●8 ●17 ●33

We can define a graded ring

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

I want to find all monomials in S of a given degree d. Can this be done in sage?

This can be done in Macaulay2 with

i1 : QQ[x, y, Degrees => {{1}, {2}}]

o1 = S

o1 : PolynomialRing

i2 : basis(4, S)

o2 = | x4 x2y y2 |

             1      3
o2 : Matrix S <--- S

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answered 2016-08-25 18:54:04 +0100

John Palmieri

8876 ●19 ●73 ●192 http://www.math.washin...

updated 2016-08-25 18:54:34 +0100

This isn't perfect, but it's close: use GradedCommutativeAlgebra. The generators in odd degrees anti-commute, so if you want an honest polynomial algebra, double the degrees of the generators:

sage: S.<x,y> = GradedCommutativeAlgebra(QQ, degrees=(2, 4))
sage: S
Graded Commutative Algebra with generators ('x', 'y') in degrees (2, 4) with relations [0] over Rational Field
sage: (x*y).degree()
6
sage: S.basis(10)
[x*y^2, x^3*y, x^5]

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answered 2016-08-26 21:28:05 +0100

done_with_fish
273 ●8 ●17 ●33

For anyone interested, here's what I came up with.

def weighted_ring_basis(n, weights):                                            
    if not 1 in weights:                                                        
        L = weights + [1]                                                       
        X = toric_varieties.WP(L)                                               
        G = X.rational_class_group()                                            
        OO = G.gens()[0]                                                        
        basis = (n*OO).lift().sections_monomials()                              
        return [f for f in basis if not f.dict().keys()[0][len(weights)]]       
    X = toric_varieties.WP(weights)                                             
    G = X.rational_class_group()                                                
    OO = G.gens()[0]                                                            
    return (n*OO).lift().sections_monomials()

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