libsingular reduces polynomials incompletely

[SimonKing]

Condensed version of a report by Oleksandr Kazymyrov ​on sage-support.

Define g by modding out in a multivariate polynomial ring:

sage: gens = 'y a0 a1 a2 b0 b1 b2 c1 c2 d0 d1 d2 d3 d4 d5 d6 d7'.split()
sage: R = PolynomialRing(GF(8), 17, gens)
sage: R.inject_variables(verbose=False)
sage: A, B, C = a0 + a1*y + a2*y^2, b0 + b1*y + b2*y^2, c1*y + c2*y^2
sage: D = d0 + d1*y + d2*y^2 + d3*y^3 + d4*y^4 + d5*y^5 + d6*y^6 + d7*y^7
sage: F = D.subs({y: B})
sage: G = A.subs({y: F}) + C
sage: g = G.mod(y^8 + y)

After modding out by a polynomial of degree 8 in y, the degree of g in y should be at most 7.

Up to Sage 9.1.rc1:

sage: g.degree(y)
14

After #27508 (Force tail reduction in polynomial quotient ring), merged in Sage 9.1.rc2:

sage: g.degree(y)
7

The problem was libsingular not doing tail reduction when calling Singular.

This ticket adds the above example as a doctest.

[jdemeyer]

I don't believe all basic arithmetic failures should be blockers, and it's dubious that this issue is even about "basic" arithmetic.

[kedlaya]

I just tried this and got an answer consistent with Singular. Maybe close this as a duplicate?

[slelievre]

The bug exists in Sage <= 9.0 but not in Sage >= 9.1.

Condensed version of the initial report, as a tentative doctest:

sage: gens = 'y a0 a1 a2 b0 b1 b2 c1 c2 d0 d1 d2 d3 d4 d5 d6 d7'.split()
sage: R = PolynomialRing(GF(8), 17, gens)
sage: R.inject_variables(verbose=False)
sage: A, B, C = a0 + a1*y + a2*y^2, b0 + b1*y + b2*y^2, c1*y + c2*y^2
sage: D = d0 + d1*y + d2*y^2 + d3*y^3 + d4*y^4 + d5*y^5 + d6*y^6 + d7*y^7
sage: F = D.subs({y: B})
sage: G = A.subs({y: F}) + C
sage: g = G.mod(y^8 + y)
sage: g.degree(y)
7

In Sage <= 9.0 we get 14 instead of 7.

If someone can suggest where to locate this doctest, I can push a branch.

Priority down from critical to major: bug fixed, only a doctest is needed.

[kedlaya]

I would suggest to put the doctest in sage/rings/polynomial/polynomial_element.pyx under mod or __mod__.

[slelievre]

Please review and add reviewer name.

Let's see if this can get in Sage 9.2.

---

New commits:

​5b3253a

Add doctest for remainder in multivariate polynomial ring

[kedlaya]

No complaints from patchbot, nor from me.

[slelievre]

Thanks!