When comparing ideals, try to avoid computing the Gröbner basis of a copy of the ideal

[SimonKing]

We define a polynomial ring, some ideal, and a copy of that ideal.

sage: P.<x,y> = QQ[]
sage: J = [x^4*y^4 + 2*x^2*y^5 + y^6 - 2/3*x^2*y^2 - 2/3*y^3 + 1/9, 9/16*y^6 - x^2*y^3 + 3/2*x*y^4 + 1/2*y^5 + 4/9*x^4 - 4/3*x^3*y + 5/9*x^2*y^2 + 2/3*x*y^3 + 1/9*y^4 + 12*y^3 - 32/3*x^2 + 16*x*y + 16/3*y^2 + 64, y^8 - 2/5*x^3*y^4 + 1/25*x^6 - 4/11*x*y^4 + 6*y^5 + 4/55*x^4 - 6/5*x^3*y + 10*y^4 - 2*x^3 + 4/121*x^2 - 12/11*x*y + 9*y^2 - 20/11*x + 30*y + 25, 1/400*x^4*y^4 - 1/5*x^5*y^2 - 1/20*x^4*y^3 + 4*x^6 + 2*x^5*y + 11/20*x^4*y^2 - 12*x^5 - 3*x^4*y + 9*x^4]*P
sage: J2 = J.gens()*P

We have:

sage: %timeit J==J2
5 loops, best of 3: 642 ms per loop
sage: _ = J.groebner_basis()
sage: %timeit J==J2
5 loops, best of 3: 642 ms per loop
sage: _ = J2.groebner_basis()
sage: %timeit J==J2
625 loops, best of 3: 6.67 µs per loop

Hence, only when the Gröbner bases of both arguments are cached, then the cached Gröbner bases are used. As one can see by reading the code, even if only one argument does not have the Gröbner basis cached, then Gröbner bases are computed for degrevlex copies of both arguments.

Why copies? We have the same ring here, and we have degrevlex anyway. So, there is no need to copy.

Suggestion: Make the algorithm slightly more clever, so that taking a copy is avoided when the ring is degrevlex anyway. One could also consider to prepend a quick test, such as: Compare the set of generators, and compute Gröbner bases only if the quick test does not suffice to prove equality.

[SimonKing]

It seems that this problem shall be dealt with at #12802. Hence, I'll mark it as duplicate.

[SimonKing]

Jeroen, why "dependency"? This here is really just a duplicate.