Trac #27508: Force tail reduction in polynomial quotient ring

I'd like to "remove squares" in some polynomials living in a polynomial
ring over `QQ`, in 2 variables: `x`,`y`. I tried to implement this by
modding out by the ideal `(x^2 - x, y^2 - y)`. Depending on the
ordering, the result of `.mod()` does not always output the polynomial I
am looking for.

Without specifying an ordering, everything seems fine:
{{{
#!python
sage: R1.<x,y> = PolynomialRing(QQ, 2)
sage: I1 = R1.ideal(["x^2 - x", "y^2 - y"])
sage: R1("x^2 + y").mod(I1)
x + y
sage: R1("x + y^2").mod(I1)
x + y
}}}

However, when specifying the order `lex` the reduction of `x + y^2` is
not as expected:
{{{
#!python
sage: R2.<x,y> = PolynomialRing(QQ, 2, order="lex")
sage: I2 = R2.ideal(["x^2 - x", "y^2 - y"])
sage: R2("x^2 + y").mod(I2)
x + y
sage: R2("x + y^2").mod(I2)
x + y^2
}}}

This issue was reported in [[https://groups.google.com/forum/#!topic
/sage-support/80Scc9pTkPM|sage-support]] where it was pointed out that
it is likely a bug in Singular, or in the Singular interface to Sage.

In particular, using the order `lex` works when
`implementation="generic"` is also specified:
{{{
#!python
sage: R3.<x,y> = PolynomialRing(QQ, 2, order="lex",
implementation="generic")
sage: I3 = R3.ideal(["x^2 - x", "y^2 - y"])
sage: R3("x^2 + y").mod(I3)
x + y
sage: R3("x + y^2").mod(I3)
x + y
}}}

For reference, I am using Sage version 8.6 on macOS Mojave 10.14.3.

PS. see also https://groups.google.com/d/msg/sage-
devel/K49-V3BbWbg/pxuoehPvAAAJ

URL: https://trac.sagemath.org/27508
Reported by: gh-rachelplayer
Ticket author(s): Dima Pasechnik
Reviewer(s): Markus Wageringel

Author: Release Manager

src/sage/libs/singular/decl.pxd

@@ -334,6 +334,7 @@ cdef extern from "singular/Singular/libsingular.h":
         bint *pairtest
         void *R
         int *S_2_R
+        bint noTailReduction
 
     ctypedef struct data_union:
         ring *uring

src/sage/libs/singular/groebner_strategy.pyx

@@ -93,6 +93,12 @@ cdef class GroebnerStrategy(SageObject):
             ...
             NotImplementedError: Only coefficient fields are implemented so far.
 
+        Check that :trac:`27508` is fixed::
+
+            sage: R2.<x,y> = PolynomialRing(QQ, 2, order="lex")
+            sage: I2 = R2.ideal(["x^2 - x", "y^2 - y"])
+            sage: R2("x^2 + y").mod(I2), R2("x + y^2").mod(I2)
+            (x + y, x + y)
         """
         if not isinstance(L, MPolynomialIdeal):
             raise TypeError("First parameter must be a multivariate polynomial ideal.")
@@ -127,6 +133,7 @@ cdef class GroebnerStrategy(SageObject):
         self._strat.sl = -1
         #- init local data struct
         initS(i, NULL, self._strat)
+        self._strat.noTailReduction = False
 
         cdef int j
         cdef bint base_ring_is_field = R.base_ring().is_field()