PS:

On 30 Jul., 12:57, Simon King <[email protected]> wrote:
> No, you are wrong. Your expected answer is definitely not correct,

But certainly there is something wrong in Singular/libsingular (which
by default is used in Sage to compute Gröbner bases) as well:

sage:
(I.groebner_basis(algorithm='toy:buchberger2')*R).interreduced_basis()
[2*x^2 + x*y, 3*x*y, 2*y^2]
sage:
(I.groebner_basis(algorithm='toy:buchberger')*R).interreduced_basis()
[2*x^2 + x*y, 3*x*y, 2*y^2]
sage:
(I.groebner_basis(algorithm='singular:std')*R).interreduced_basis()
[x^2*y, x*y^2, 2*x^2 + x*y, 3*x*y, 2*y^2]
sage:
(I.groebner_basis(algorithm='libsingular:std')*R).interreduced_basis()
[x^2*y, x*y^2, 2*x^2 + x*y, 3*x*y, 2*y^2]
sage:
(I.groebner_basis(algorithm='libsingular:slimgb')*R).interreduced_basis()
[2*x^2 + x*y, 3*x*y, 2*y^2]

So, it looks like both toy:buchberger and toy:buchberger2 and slimgb
coincide - and they are in majority. But again, they are wrong:

sage: (y*I.0 -2*y^3*I.1 -x*I.
2).reduce(I.groebner_basis(algorithm='libsingular:slimgb'))
x*y^2

So, this will actually be several bug reports to Singular.

Thank you for pointing us to this!
Simon

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