reduce in quotient rings is broken

[yyyyx4]

Sage 10.3.beta8:

sage: R.<x,y> = GF(11)[]
sage: I = [x^2+1, x+y]
sage: J = [x]
sage: 1 in R.ideal(I + J)
True
sage: 1 in R.quotient(I).ideal(J)
False

See also #33217.

[yyyyx4]

@mwageringel This seems related to some code you touched in #33217; could you maybe have a look? I don't know much about this topic.

[mwageringel]

Uh oh, the problem is that the quotient ideal I is not a Gröbner basis, but that is required by the Singular backend:

4.19.1 qring
SINGULAR offers the opportunity to calculate in quotient rings (factor rings), i.e., rings modulo an ideal. The ideal has to be given as a standard basis.

Indeed:

sage: 1 in R.quotient(R.ideal(I).groebner_basis()).ideal(J)
True   # correct

The relevant implementation is

sage/src/sage/rings/polynomial/multi_polynomial_ideal.py

Lines 5648 to 5655 in 30b3d78

	Q = self.ring()
	gb = self.groebner_basis()
	# In quotient rings, gb is not a Gröbner basis of self, but gb0 is
	# a (possibly non-reduced) Gröbner basis of the preimage of self in
	# the cover ring (see :trac:`33217`). We only use Gröbner bases of
	# pre-existing ideals to potentially take advantage of caching.
	gb0 = Q.defining_ideal().groebner_basis() + [g.lift() for g in gb]
	f0 = f.lift().reduce(gb0)

where already line 5649 does not yield a meaningful result in this case.

I do not see a simple fix other than to throw an error if the defining ideal of the quotient ring is not already a Gröbner basis. Otherwise, one would have to replace the quotient ring by one that is defined in terms of a Gröbner basis, but that would be an awkward thing to do in the reduce method unless some caching mechanism is used.

[yyyyx4]

Why can't it just compute a Gröbner basis on demand when needed and cache it after the first time? I was under the impression that this is how things are done all over the multivariate codebase, is it not?

[maxale]

Issue #33982 may be related. Both issues may have caused by buggy implementation of ideals over non-fields.