Hi!
In the past I intensely used pexpect interfaces to GAP and even more to
Singular. Recently I switched to using libgap, which turned out to be a
good idea. Now I consider to replace "singular-via-pexpect" by
libsingular. But I got stuck very early.
Setting: I am dealing with graded-homogeneous ideals over
graded-commutative rings (i.e., even degree elements are in the centre,
whereas odd-degree elements anti-commute with each other).
The term order is given by a matrix. I need to compute Gröbner bases of
such ideals, and for efficiency I am often imposing a degree bound.
Also, I am using Singular's std() command in the form std(I,J), which
computes a Gröbner basis of the ideal generated by I and J under the
assumption that I is a Gröbner basis.
A graded-commutative ring can easily be implemented using Singular's
"super-commutative" rings. Ostensibly, they are wrapped in
sage.rings.polynomial.plural.SCA.
However:
sage: from sage.rings.polynomial.plural import SCA
sage: S = SCA(GF(3), ('x1','x2','x3','x4','x5'), [0,1,2])
sage: I = S*S.gens()*S
sage: type(I)
<class 'sage.rings.noncommutative_ideals.Ideal_nc'>
sage: hasattr(I, 'groebner_basis')
False
sage: groebner = sage.libs.singular.function_factory.ff.groebner
sage: groebner(I, ring=S)
Traceback (most recent call last):
...
TypeError: Cannot call Singular function 'groebner' with ring
parameter of type '<class
'sage.rings.quotient_ring.QuotientRing_nc_with_category'>'
I thought that a couple of years ago a Singular (or rather Plural) wrapper
was created for ideals over non-commutative rings. Indeed, Singular's
G-algebras and their ideals are wrapped to the extent that Gröbner bases
can be computed. However, Gröbner basis computation over
super-commutative algebras is a lot more efficient than over general
G-algebras, if I recall correctly. On the other hand, I think I have
been told at some point that if one creates a G-algebra that is in fact
super-commutative, then Singular will notice and will automatically
apply the more efficient specialised algorithms --- but I am not sure if
I reacall that correctly.
Questions:
Is it true that Singular will automatically recognise whether a
G-algebra is super-commutative?
If yes: How to create a G-algebra whose term order is defined by a
given matrix? How to then use std(I,J), with degree bound?
If no: Is everything already available in Sage via libsingular? If yes:
How? If no: Is there a ticket for it?
Best regards,
Simon
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