Trac #12529: libsingular reduces polynomials incompletely

Condensed version of a report by Oleksandr Kazymyrov
[http://groups.google.com/group/sage-
support/browse_thread/thread/b90fbb622ddb55ab 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.

URL: https://trac.sagemath.org/12529
Reported by: SimonKing
Ticket author(s): Samuel Lelièvre
Reviewer(s): Kiran Kedlaya

Author: Release Manager

src/sage/rings/polynomial/polynomial_element.pyx

@@ -2788,6 +2788,21 @@ cdef class Polynomial(CommutativeAlgebraElement):
             -1
             sage: (x^3 + x - 1) % (x^2 - 1)
             2*x - 1
+
+        TESTS:
+
+        Check the problem reported at :trac:`12529` is fixed::
+
+            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
         """
         return self % other