Trac #21884: giacpy_sage doctest failure and new libsingular cf.type == n_unknown ring test

Related to #21860 : one failure in ptestlong :

{{{
charpent@asus16-ec:/usr/local/sage-7$ sage -t --long --warn-long 100.3
src/sage/rings/polynomial/multi_polynomial_ideal.py
Running doctests with ID 2016-11-17-09-17-43-cce569e2.
Git branch: t/21860/giacblas
Using --optional=database_gap,giac,giacpy_sage,mpir,python2,sage
Doctesting 1 file.
sage -t --long --warn-long 100.3
src/sage/rings/polynomial/multi_polynomial_ideal.py
**********************************************************************
File "src/sage/rings/polynomial/multi_polynomial_ideal.py", line 3533,
in sage.rings.polynomial.multi_polynomial_ideal.NCPolynomialIdeal.groebn
er_basis
Failed example:
    ideal(J.transformed_basis()).change_ring(P).interreduced_basis()  #
optional - giacpy_sage
Expected:
    [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c,
c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
Got:
    [7*a - 420*c^3 + 158*c^2 + 8*c - 7, 7*b + 210*c^3 - 79*c^2 + 3*c,
84*c^4 - 40*c^3 + c^2 + c]
**********************************************************************
1 item had failures:
   1 of  67 in sage.rings.polynomial.multi_polynomial_ideal.NCPolynomial
Ideal.groebner_basis
    [723 tests, 1 failure, 9.10 s]
----------------------------------------------------------------------
sage -t --long --warn-long 100.3
src/sage/rings/polynomial/multi_polynomial_ideal.py  # 1 doctest failed
----------------------------------------------------------------------
Total time for all tests: 9.2 seconds
    cpu time: 15.5 seconds
    cumulative wall time: 9.1 seconds
}}}

in fact it is related to the singular update #17254 where all the
{{{
if r.ringtype == 0:
}}}
were replaced by
{{{
if r.cf.type == n_unknown:
}}}
but it seems not equivalent with QQ coefficients.

Remarks: The interred_libsingular function of
multi_polynomial_ideal_libsingular.pyx
ends with an explicit:
{{{
# divide head by coeffi
...
}}}
and the groebner_basis function doc have still five examples giving:
{{{
[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4
- 10/21*c^3 + 1/84*c^2 + 1/84*c]
}}}
so what is the best thing for the interreduced_basis command to output?

URL: https://trac.sagemath.org/21884
Reported by: charpent
Ticket author(s): Frederic Han
Reviewer(s): Emmanuel Charpentier

Author: Release Manager

src/sage/rings/polynomial/multi_polynomial_ideal.py

@@ -3530,7 +3530,8 @@ def groebner_basis(self, algorithm='', deg_bound=None, mult_bound=None, prot=Fal
             sage: gb  # optional - giacpy_sage
             [c^3 - 79/210*c^2 + 1/30*b + 1/70*c, b^2 - 3/5*c^2 - 1/5*b + 1/5*c, b*c + 6/5*c^2 - 1/10*b - 2/5*c, a + 2*b + 2*c - 1]
 
-            sage: ideal(J.transformed_basis()).change_ring(P).interreduced_basis()  # optional - giacpy_sage
+            sage: J.groebner_basis.set_cache(gb)  # optional - giacpy_sage
+            sage: ideal(J.transformed_basis()).change_ring(P).interreduced_basis()  # testing trac 21884
             [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
 
         Giac's gbasis over `\QQ` can benefit from a probabilistic lifting and

src/sage/rings/polynomial/multi_polynomial_ideal_libsingular.pyx

@@ -280,7 +280,7 @@ def interred_libsingular(I):
         sage: P.<x,y,z> = PolynomialRing(QQ)
         sage: I = ideal( x^2 - 3*y, y^3 - x*y, z^3 - x, x^4 - y*z + 1 )
         sage: I.interreduced_basis()
-        [y*z^2 - 81*x*y - 9*y - z, z^3 - x, x^2 - 3*y, 9*y^2 - y*z + 1]
+        [y*z^2 - 81*x*y - 9*y - z, z^3 - x, x^2 - 3*y, y^2 - 1/9*y*z + 1/9]
     """
     global singular_options
 
@@ -310,7 +310,7 @@ def interred_libsingular(I):
 
 
     # divide head by coefficients
-    if r.cf.type == n_unknown:
+    if r.cf.type != n_Z and r.cf.type != n_Znm and r.cf.type != n_Zn and r.cf.type != n_Z2m :
         for j from 0 <= j < IDELEMS(result):
             p = result.m[j]
             if p: