Trac #19423: Merge #19946 to fix merge conflict

src/sage/rings/asymptotic/asymptotic_ring.py

@@ -283,6 +283,36 @@
     sage: (1 + 1/z + O(1/z^5))^(1 + 1/z)
     1 + z^(-1) + z^(-2) + 1/2*z^(-3) + 1/3*z^(-4) + O(z^(-5))
 
+.. NOTE::
+
+    In the asymptotic ring
+    ::
+
+        sage: M.<n> = AsymptoticRing(growth_group='QQ^n * n^QQ', coefficient_ring=ZZ)
+
+    the operation
+    ::
+
+        sage: (1/2)^n
+        Traceback (most recent call last):
+        ...
+        ValueError: 1/2 is not in Exact Term Monoid QQ^n * n^QQ
+        with coefficients in Integer Ring. ...
+
+    fails, since the rational `1/2` is not contained in `M`. You can use
+    ::
+
+        sage: n.rpow(1/2)
+        (1/2)^n
+
+    instead. (See also the examples in
+    :meth:`ExactTerm.rpow() <sage.rings.asymptotic.term_monoid.ExactTerm.rpow>`
+    for a detailed explanation.)
+    Another way is to use a larger coefficent ring::
+
+        sage: M_QQ.<n> = AsymptoticRing(growth_group='QQ^n * n^QQ', coefficient_ring=QQ)
+        sage: (1/2)^n
+        (1/2)^n
 
 Multivariate Arithmetic
 ^^^^^^^^^^^^^^^^^^^^^^^
@@ -1391,6 +1421,18 @@ def __pow__(self, exponent, precision=None):
             ValueError: Cannot take s + t to the exponent s.
             > *previous* ValueError: Cannot determine main term of s + t
             since there are several maximal elements s, t.
+
+        Check that :trac:`19946` is fixed::
+
+            sage: A.<n> = AsymptoticRing('QQ^n * n^QQ', SR)
+            sage: e = 2^n; e
+            2^n
+            sage: e.parent()
+            Asymptotic Ring <SR^n * n^SR> over Symbolic Ring
+            sage: e = A(e); e
+            2^n
+            sage: e.parent()
+            Asymptotic Ring <QQ^n * n^QQ> over Symbolic Ring
         """
         if not self.summands:
             if exponent == 0:
@@ -1632,6 +1674,14 @@ def rpow(self, base, precision=None):
             ArithmeticError: Cannot construct y^x in Growth Group x^ZZ
             > *previous* TypeError: unsupported operand parent(s) for '*':
             'Growth Group x^ZZ' and 'Growth Group SR^x'
+
+        Check that :trac:`19946` is fixed::
+
+            sage: A.<n> = AsymptoticRing('QQ^n * n^QQ', SR)
+            sage: n.rpow(2)
+            2^n
+            sage: _.parent()
+            Asymptotic Ring <QQ^n * n^SR> over Symbolic Ring
         """
         if isinstance(base, AsymptoticExpansion):
             return base.__pow__(self, precision=precision)

src/sage/rings/asymptotic/growth_group.py

@@ -873,6 +873,14 @@ def _rpow_(self, base):
         sage: x = G('x')
         sage: (x * log(x)).rpow(2)  # indirect doctest
         2^(x*log(x))
+
+    ::
+
+        sage: n = GrowthGroup('QQ^n * n^QQ')('n')
+        sage: n.rpow(2)
+        2^n
+        sage: _.parent()
+        Growth Group QQ^n * n^QQ
     """
     if base == 0:
         raise ValueError('%s is not an allowed base for calculating the '

src/sage/rings/asymptotic/growth_group_cartesian.py

@@ -692,11 +692,17 @@ def _pushout_(self, other):
             Growth Group QQ^x * x^QQ
             sage: cm.common_parent(GrowthGroup('QQ^x * x^ZZ'), GrowthGroup('ZZ^x * x^QQ'))
             Growth Group QQ^x * x^QQ
+
+        ::
+
+            sage: pushout(GrowthGroup('QQ^n * n^QQ'), GrowthGroup('SR^n'))
+            Growth Group SR^n * n^QQ
         """
         from growth_group import GenericGrowthGroup, AbstractGrowthGroupFunctor
         from misc import merge_overlapping
         from misc import underlying_class
 
+        Sfactors = self.cartesian_factors()
         if isinstance(other, GenericProduct):
             Ofactors = other.cartesian_factors()
         elif isinstance(other, GenericGrowthGroup):
@@ -759,7 +765,7 @@ def next(self):
                     self.factors = tuple()
 
         from itertools import groupby
-        S = it(groupby(self.cartesian_factors(), key=lambda k: k.variable_names()))
+        S = it(groupby(Sfactors, key=lambda k: k.variable_names()))
         O = it(groupby(Ofactors, key=lambda k: k.variable_names()))
 
         newS = []
@@ -786,9 +792,9 @@ def next(self):
 
         assert(len(newS) == len(newO))
 
-        if (len(self.cartesian_factors()) == len(newS) and
-            len(other.cartesian_factors()) == len(newO)):
-            # We had already all factors in each of the self and
+        if (len(Sfactors) == len(newS) and
+            len(Ofactors) == len(newO)):
+            # We had already all factors in each of self and
             # other, thus splitting it in subproblems (one for
             # each factor) is the strategy to use. If a pushout is
             # possible :func:`sage.categories.pushout.pushout`

src/sage/rings/asymptotic/term_monoid.py

@@ -3308,6 +3308,32 @@ def rpow(self, base):
             Growth Group QQ^x * x^ZZ * log(x)^ZZ
             > *previous* TypeError: unsupported operand parent(s) for '*':
             'Growth Group QQ^x * x^ZZ * log(x)^ZZ' and 'Growth Group ZZ^(x^2)'
+
+        ::
+
+            sage: T = TermMonoid('exact', GrowthGroup('QQ^n * n^QQ'), SR)
+            sage: n = T('n')
+            sage: n.rpow(2)
+            2^n
+            sage: _.parent()
+            Exact Term Monoid QQ^n * n^SR with coefficients in Symbolic Ring
+
+        Above, we get ``QQ^n * n^SR``. The reason is the following:
+        Since $n = 1_{SR} \cdot (1_{\QQ})^n \cdot n^{1_{\QQ}}$, we have
+
+        .. MATH::
+
+            2^n = (2_{\QQ})^{1_{SR} \cdot (1_{\QQ})^n \cdot n^{1_{\QQ}}}
+            = \left( (2_{\QQ})^n \cdot n^{0_{\QQ}} \right)^{1_{SR}}
+            = \left((2_{\QQ})^{1_{SR}}\right)^n \cdot n^{0_{\QQ} 1_{SR}}
+            = (2_{\QQ})^n \cdot n^{0_{SR}}
+
+        where ::
+
+            sage: (QQ(2)^SR(1)).parent(), (QQ(0)*SR(1)).parent()
+            (Rational Field, Symbolic Ring)
+
+        was used.
         """
         P = self.parent()