Trac #20062: Make _floordiv_() return the Euclidean quotient for power series over fields

There exists a method `PowerSeries_poly.__floordiv__()`, but it is not
clear how it differs from ordinary division (see
comment:43:ticket:15601), or how it should differ mathematically.

We replace this method by a new method `PowerSeries._floordiv_()`, which
returns the Euclidean quotient over fields and is a deprecated alias for
`_div_()` over other rings.

URL: http://trac.sagemath.org/20062
Reported by: pbruin
Ticket author(s): Peter Bruin
Reviewer(s): Bruno Grenet

Author: Release Manager

src/sage/rings/power_series_poly.pyx

@@ -10,8 +10,6 @@ from power_series_ring_element cimport PowerSeries
 from sage.structure.element cimport Element, ModuleElement, RingElement
 from infinity import infinity, is_Infinite
 from sage.libs.all import PariError
-from power_series_ring_element import is_PowerSeries
-import rational_field
 from sage.misc.superseded import deprecated_function_alias
 
 cdef class PowerSeries_poly(PowerSeries):
@@ -557,34 +555,6 @@ cdef class PowerSeries_poly(PowerSeries):
         """
         return PowerSeries_poly(self._parent, c * self.__f, self._prec, check=False)
 
-    def __floordiv__(self, denom):
-        """
-        EXAMPLES::
-
-            sage: R.<t> = ZZ[[]] ; f = t**10-1 ; g = 1+t+t^7 ; h = f.add_bigoh(20)
-            sage: f // g
-            -1 + t - t^2 + t^3 - t^4 + t^5 - t^6 + 2*t^7 - 3*t^8 + 4*t^9 - 4*t^10 + 5*t^11 - 6*t^12 + 7*t^13 - 9*t^14 + 12*t^15 - 16*t^16 + 20*t^17 - 25*t^18 + 31*t^19 + O(t^20)
-            sage: (f // g) * g
-            -1 + t^10 + O(t^20)
-            sage: g // h
-            -1 - t - t^7 - t^10 - t^11 - t^17 + O(t^20)
-            sage: (g // h) * h
-            1 + t + t^7 + O(t^20)
-            sage: h // g
-            -1 + t - t^2 + t^3 - t^4 + t^5 - t^6 + 2*t^7 - 3*t^8 + 4*t^9 - 4*t^10 + 5*t^11 - 6*t^12 + 7*t^13 - 9*t^14 + 12*t^15 - 16*t^16 + 20*t^17 - 25*t^18 + 31*t^19 + O(t^20)
-            sage: (h // g) * g
-            -1 + t^10 + O(t^20)
-        """
-        try:
-            return PowerSeries.__div__(self, denom)
-        except (PariError, ZeroDivisionError) as e: # PariError to general?
-            if is_PowerSeries(denom) and denom.degree() == 0 and denom[0] in self._parent.base_ring():
-                denom = denom[0]
-            elif not denom in self._parent.base_ring():
-                raise ZeroDivisionError, e
-            return PowerSeries_poly(self._parent,
-                                             self.__f // denom, self._prec)
-
     def __lshift__(PowerSeries_poly self, n):
         """
         Shift self to the left by n, i.e. multiply by x^n.

src/sage/rings/power_series_ring_element.pyx

@@ -108,7 +108,8 @@ import rational_field, integer_ring
 from integer import Integer
 from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
 from sage.libs.pari.all import pari
-from sage.misc.superseded import deprecated_function_alias
+from sage.misc.superseded import deprecated_function_alias, deprecation
+from warnings import warn
 
 from sage.categories.fields import Fields
 _Fields = Fields()
@@ -1039,6 +1040,32 @@ cdef class PowerSeries(AlgebraElement):
             num = self
         return num*inv
 
+    cpdef RingElement _floordiv_(self, RingElement denom):
+        """
+        Euclidean division (over fields) or ordinary division (over
+        other rings; deprecated).
+
+        EXAMPLES::
+
+            sage: A.<q> = GF(7)[[]]
+            sage: (q^2 - 1) // (q + 1)
+            doctest:...: UserWarning: the operator // now returns the Euclidean quotient for power series over fields, use / for the true quotient
+            6 + q + O(q^20)
+
+            sage: R.<t> = ZZ[[]]
+            sage: (t**10 - 1) // (1 + t + t^7)
+            doctest:...: DeprecationWarning: the operator // is deprecated for power series over non-fields, use / instead
+            See http://trac.sagemath.org/20062 for details.
+            -1 + t - t^2 + t^3 - t^4 + t^5 - t^6 + 2*t^7 - 3*t^8 + 4*t^9 - 4*t^10 + 5*t^11 - 6*t^12 + 7*t^13 - 9*t^14 + 12*t^15 - 16*t^16 + 20*t^17 - 25*t^18 + 31*t^19 + O(t^20)
+        """
+        try:
+            q, r = self.quo_rem(denom)
+            warn("the operator // now returns the Euclidean quotient for power series over fields, use / for the true quotient")
+            return q
+        except (AttributeError, NotImplementedError):
+            deprecation(20062, "the operator // is deprecated for power series over non-fields, use / instead")
+            return self._div_(denom)
+
     def __mod__(self, other):
         """
         EXAMPLES::