Completion of polynomial rings and their fraction fields

src/sage/categories/pushout.py

@@ -2466,12 +2466,12 @@ class CompletionFunctor(ConstructionFunctor):
         True
 
         sage: P.<x> = ZZ[]
-        sage: Px = P.completion(x) # currently the only implemented completion of P
+        sage: Px = P.completion(x)
         sage: Px
-        Power Series Ring in x over Integer Ring
+        Completion of Univariate Polynomial Ring in x over Integer Ring at x
         sage: F3 = Px.construction()[0]
         sage: F3(GF(3)['x'])
-        Power Series Ring in x over Finite Field of size 3
+        Completion of Univariate Polynomial Ring in x over Finite Field of size 3 at x
 
     TESTS::
 
@@ -2484,7 +2484,7 @@ class CompletionFunctor(ConstructionFunctor):
         (1 + O(5^20))*a + 3 + 2*5 + 2*5^2 + 2*5^3 + 2*5^4 + 2*5^5 + 2*5^6 + 2*5^7 + 2*5^8 + 2*5^9 + 2*5^10 + 2*5^11 + 2*5^12 + 2*5^13 + 2*5^14 + 2*5^15 + 2*5^16 + 2*5^17 + 2*5^18 + 2*5^19 + O(5^20)
     """
     rank = 4
-    _real_types = ['Interval', 'Ball', 'MPFR', 'RDF', 'RLF', 'RR']
+    _real_types = [None, 'Interval', 'Ball', 'MPFR', 'RDF', 'RLF', 'RR']
     _dvr_types = [None, 'fixed-mod', 'floating-point', 'capped-abs', 'capped-rel', 'lattice-cap', 'lattice-float', 'relaxed']
 
     def __init__(self, p, prec, extras=None):
@@ -2541,7 +2541,7 @@ def __init__(self, p, prec, extras=None):
             from sage.rings.infinity import Infinity
             if self.p == Infinity:
                 if self.type not in self._real_types:
-                    raise ValueError("completion type must be one of %s" % (", ".join(self._real_types)))
+                    raise ValueError("completion type must be one of %s" % (", ".join(self._real_types[1:])))
             elif self.type not in self._dvr_types:
                 raise ValueError("completion type must be one of %s" % (", ".join(self._dvr_types[1:])))
 
@@ -2777,9 +2777,9 @@ def commutes(self, other):
         functors in opposite order works. It does::
 
             sage: P.<x> = ZZ[]
-            sage: C = P.completion(x).construction()[0]
+            sage: C = P.completion('x').construction()[0]
             sage: R = FractionField(P)
-            sage: hasattr(R,'completion')
+            sage: hasattr(R, 'completion')
             False
             sage: C(R) is Frac(C(P))
             True

src/sage/monoids/trace_monoid.py

@@ -864,7 +864,7 @@ def number_of_words(self, length):
             sage: M.number_of_words(3)                                                  # needs sage.graphs
             48
         """
-        psr = PowerSeriesRing(ZZ, default_prec=length + 1)
+        psr = PowerSeriesRing(ZZ, 'x', default_prec=length + 1)
         return psr(self.dependence_polynomial()).coefficients()[length]
 
     @cached_method

src/sage/rings/big_oh.py

@@ -13,6 +13,8 @@
 
 lazy_import('sage.rings.padics.factory', ['Qp', 'Zp'])
 from sage.rings.polynomial.polynomial_element import Polynomial
+from sage.rings.fraction_field_element import FractionFieldElement
+from sage.rings.fraction_field_FpT import FpTElement
 
 try:
     from .puiseux_series_ring_element import PuiseuxSeries
@@ -47,6 +49,16 @@ def O(*x, **kwds):
         O(x^100)
         sage: 1/(1+x+O(x^5))
         1 - x + x^2 - x^3 + x^4 + O(x^5)
+
+    Completion at other places also works::
+
+        sage: x^3 + O((x^2 + 1)^10)
+        -x + x*(x^2 + 1) + O((x^2 + 1)^10)
+        sage: x^3 + O(1/x^10)  # completion at infinity
+        x^3 + O(x^-10)
+
+    An example with several variables::
+
         sage: R.<u,v> = QQ[[]]
         sage: 1 + u + v^2 + O(u, v)^5
         1 + u + v^2 + O(u, v)^5
@@ -126,11 +138,6 @@ def O(*x, **kwds):
 
     ::
 
-        sage: R.<x> = QQ[]
-        sage: O(2*x)
-        Traceback (most recent call last):
-        ...
-        NotImplementedError: completion only currently defined for the maximal ideal (x)
         sage: R.<x> = LazyPowerSeriesRing(QQ)
         sage: O(x^5)
         O(x^5)
@@ -169,13 +176,29 @@ def O(*x, **kwds):
     if isinstance(x, power_series_ring_element.PowerSeries):
         return x.parent()(0, x.degree(), **kwds)
 
+    if isinstance(x, (FractionFieldElement, FpTElement)):
+        if x.denominator().is_one():
+            x = x.numerator()
+        elif x.numerator().is_one():
+            x = x.denominator()
+            if isinstance(x, Polynomial) and x.is_monomial():
+                from sage.rings.infinity import infinity
+                C = x.parent().completion(infinity)
+                n = x.degree()
+                return C.zero().add_bigoh(n)
+
     if isinstance(x, Polynomial):
-        if x.parent().ngens() != 1:
+        A = x.parent()
+        if A.ngens() != 1:
             raise NotImplementedError("completion only currently defined "
                                       "for univariate polynomials")
+        if x.is_monomial():
+            C = A.completion(A.variable_name())
+            n = x.degree()
         if not x.is_monomial():
-            raise NotImplementedError("completion only currently defined "
-                                      "for the maximal ideal (x)")
+            p, n = x.perfect_power()
+            C = A.completion(p)
+        return C.zero().add_bigoh(n)
 
     if isinstance(x, (int, Integer, Rational)):
         # p-adic number