Dear Sage team,
I often have to deal with graded commutative algebras, which I emulate
in Singular using SuperCommutative plus a weighted monomial ordering.
I also have weighted homogeneous ideals, and I want to compute the
Hilbert function.
Problem 1:
Singular can compute the Hilbert function only for commutative rings.
Problem 2:
I have one commutative example where Singular apparently has an
integer overflow.
The nasty thing is that Singular does not complain but simply returns
a wrong answer
(fortunately I knew the result from other sources).
Now I see that MPolynomialRing ideals in Sage also have a method
hilbert_series. The doc-string says:
Let I = self be a homogeneous ideal and R =
self.ring() be a graded commutative algebra (R =
oplus R_d) over a field K. Then the Hilbert function is...
I was very surprised to see "graded commutative algebra" being
mentioned. But how can one create them? Up to now I thought that
generators of polynomial rings in Sage have degree 1, and there is no
way to change it. And now even "graded commutative" (hence, in
general, NON-commutative)?
The index of the Sage reference manual contains the word "graded" only
once (in a different context), so, it didn't help.
Can you help? And would the hilbert_function method avoid integer
overflow?
Yours sincerely
Simon
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