Trac #17638: Division of polynomials produces errors when using local…

… orderings

Consider two polynomials f,g in a ring define using a local ordering,
e.g.,
{{{
R.<x,y>=PolynomialRing(QQ,order='neglex')
}}}
If the leading monomial of g is 1 then f/g will produces a wrong result,
namely f divided by the independent coefficient of g. Note that in that
case g.is_unit() yields True, since for Singular R is not Q[x,y] but the
ordering localization (i.e., one can divide by the polynomials whose
leading monomial is 1).

PS: It is my first ticket, I apologize for the mistakes.

URL: https://trac.sagemath.org/17638
Reported by: enriqueartal
Ticket author(s): Mckenzie West
Reviewer(s): Luis Felipe Tabera Alonso, Thierry Monteil

src/sage/rings/polynomial/multi_polynomial_libsingular.pyx

@@ -2315,6 +2315,15 @@ cdef class MPolynomial_libsingular(MPolynomial):
             Traceback (most recent call last):
             ...
             ZeroDivisionError: rational division by zero
+
+        Ensure that :trac:`17638` is fixed::
+
+            sage: R.<x,y> = PolynomialRing(QQ, order="neglex")
+            sage: f = 1 + y
+            sage: g = 1 + x
+            sage: h = f/g
+            sage: h*g == f
+            True
         """
         cdef poly *p
         cdef MPolynomial_libsingular right = <MPolynomial_libsingular>right_ringelement