reduced Groebner basis not unique

[mariah]

Using sage-4.8

sage: R.<x,y> = PolynomialRing(ZZ)
sage: I = R*[x^2-y, 2*y]
sage: J = R*[x^2+y, 2*y]

I and J are visibly the same ideal, but Sage finds different reduced Groebner bases:

sage: I.groebner_basis()
[x^2 - y, 2*y]
sage: J.groebner_basis()
[x^2 + y, 2*y]
sage: I == J   // should say True
False

[john_perry]

Correct me if I am wrong, but you cannot have a reduced Groebner basis over a ring that is not a field. Besides, the ideals are not the same, even if their varieties are.

Singular seems to feel this way:

• Singular is computing the basis.
• According to Remark 1.6.14 in A Singular Introduction to Commutative Algebra, if you want to compute a standard basis over a ring which is merely Noetherian (not necessarily a field, as in Definition 1.6.1), you need to have agreement of leading terms (which includes coefficients), not leading monomials.
• See this answer in ​the Singular forums.

Macaulay also feels this way:

• I installed Macaulay2, computed groebner bases for both I and J, and got the same thing singular computes.
• Macaulay2's ​webpage implies the same.

Unless I'm wrong, this is not a bug.

Edit: I had ring and field switched in the first sentence.

[john_perry]

Replying to john_perry:

Besides, the ideals are not the same, even if their varieties are.

This was a dumb thing to say, & I had doubts almost immediately after hitting the Submit button. In fact, Macaulay2 recognizes I==J, even though it computes different gb's. Singular does not recognize this.

[john_perry]

I think I see a way to get this to work.

The first thing that can be tried is whether the groebner bases are equal, which is what we are doing now. If that succeeds, then great.

Otherwise, we can compare by reducing the elements of one groebner basis over the other's groebner basis. If all reductions give us 0, then we return true. Otherwise, we return false.

I think this is related to #12802, and this trick should fix both of them: __lt__ can test if the first is contained in the second, __gt__ can check if the second is contained in the first, and __eq__ checks if both are satisfied.

This would be easy to implement, but is the algorithm I'm outlining correct?

[john_perry]

Hello

I'm still of the opinion that what I wrote about reduced Groebner bases in integer rings is correct, given the behavior of Singular and Macaulay2.

That said, the incorrect conclusion that I != J is easily fixed, using the algorithm I outlined. I have uploaded a patch to #12802 that does precisely this.

Assuming that what I've done there is correct, is there a way to mark this patch as a duplicate, or something similar?

[was]

john_perry: to mark this as a dup:

1. Set yourself to reviewer
2. give it positive review
3. change the milestone to sage-duplicate

William

[john_perry]

Replying to was:

john_perry: to mark this as a dup:

Thx. Since testing equality of ideals is equivalent to testing containment both ways, that's what I'll do.