Is there an implementation of such a thing as in the title?
I am facing the following design problem, and I'd love to get some pointers. I
have a category of parents that are infinite dimensional super algebras but are
graded with finite dimensional graded pieces. For the sake of clarity think of
a polynomial differential algebra that is QQ[x_1,x_2,x_3,...,y_1,y_2,y_3,....]
where the variables x_i and y_i have degree i. I can work with ideals and
quotients of these guys by computing the quotient degree by degree.
Now the issue is to compute the graded dimension of the quotient as a formal
power series. The way I am implementing this is by computing a basis of the
degree part n of the algebra and then the degree part n of the ideal.
In the particular case of the polynomial algebra above, I have another
implementation: if I only care about the hilbert series up to degree n, then I
work in the quotient by (x_j, y_j) for j>n, this gives me a finite type
polynomial algebra and I can ask for a grobner basis of the ideal before
computing the hilbert series. This is several orders of magnitude faster.
Now my question: in my setup I need some of the x's or y's to be odd,
anticommutative variables. And even worse their weights might be rational
numbers and not only integers. Buchberger's algorithm works with minimal
variations in the super-commutative case. If I disregard the issue of rational
degrees, is there an implementation of these "grobner bases" for super
commutative algebras in Sage?
Best,
R.
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