Singular `groebner_basis(deg_bound=1)` returns conflicting groebner basis including $1$ in the basis.

[Georgi Guninski]

Singular `groebner_basis(deg_bound=1)` returns conflicting groebner
basis including $1$ in the basis.

sage: K.<x,y>=GF(2)[];I=[x*y,(x+1)*y,(y+1)*x,(y+1)*(x+1)]
sage: gb=Ideal(I).groebner_basis(deg_bound=1,algorithm="singular:groebner");gb
[x*y, x, y, 1]
sage: gb=Ideal(I).groebner_basis(deg_bound=1,algorithm="singular:std");gb
[x*y, x, y, 1]
sage: gb=Ideal(I).groebner_basis(deg_bound=None,algorithm="singular:std");gb
[1]

[Georgi Guninski]

I have other testcases, where `deg_bound=2` (and possibly greater)
fails in the same way.
This sucks much, can anyone else confirm it?

[David Roe]

I think that you're telling the Gröbner basis algorithm to ignore the xy term, so it can't reduce everything.  If you give a degree bound, I don't think you're guaranteed to get an actual Gröbner basis, and I think this is just a manifestation of that.  The ideal you give is the unit ideal, so it's not surprising that 1 is in the basis: the correct answer is [1].

David

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[Travis Scrimshaw]

To me, they both look like Gröbner bases (which are not unique), but only [1] is the reduced Gröbner basis (which are unique). It’s analogous to echelon forms versus the RREF of a matrix. I don’t think there is anything wrong here.

Best,

Travis