Implement QQbar^QQ as action

[burcin]

This ticket implements powering in QQbar and AA by QQ exponents as an action.

This will fix the following two bugs:

Reported by Alex Raichev on sage-support:

sage: n = SR.var('n'); QQbar(2)^n
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-6-01e1c4db05c9> in <module>()
----> 1 n = SR.var('n'); QQbar(Integer(2))**n

/usr/local/src/sage-config/local/lib/python2.7/site-packages/sage/rings/qqbar.pyc in __pow__(self, e)
   4106         if self == self.parent().one():
   4107             return self.parent().one()
-> 4108         e = QQ._coerce_(e)
   4109         n = e.numerator()
   4110         d = e.denominator()

/usr/local/src/sage-config/src/sage/structure/parent_old.pyx in sage.structure.parent_old.Parent._coerce_ (build/cythonized/sage/structure/parent_old.c:5673)()
    227     def _coerce_(self, x):            # Call this from Python (do not override!)
    228         if self._element_constructor is not None:
--> 229             return self.coerce(x)
    230         check_old_coerce(self)
    231         return self._coerce_c(x)

/usr/local/src/sage-config/src/sage/structure/parent.pyx in sage.structure.parent.Parent.coerce (build/cythonized/sage/structure/parent.c:10829)()
   1157                 except Exception:
   1158                     _record_exception()
-> 1159             raise TypeError("no canonical coercion from %s to %s" % (parent(x), self))
   1160         else:
   1161             return (<map.Map>mor)._call_(x)

TypeError: no canonical coercion from Symbolic Ring to Rational Field

sage: QQbar(2)^1.0
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-7-aba7adf5dc93> in <module>()
----> 1 QQbar(Integer(2))**RealNumber('1.0')

/usr/local/src/sage-config/local/lib/python2.7/site-packages/sage/rings/qqbar.pyc in __pow__(self, e)
   4106         if self == self.parent().one():
   4107             return self.parent().one()
-> 4108         e = QQ._coerce_(e)
   4109         n = e.numerator()
   4110         d = e.denominator()

/usr/local/src/sage-config/src/sage/structure/parent_old.pyx in sage.structure.parent_old.Parent._coerce_ (build/cythonized/sage/structure/parent_old.c:5673)()
    227     def _coerce_(self, x):            # Call this from Python (do not override!)
    228         if self._element_constructor is not None:
--> 229             return self.coerce(x)
    230         check_old_coerce(self)
    231         return self._coerce_c(x)

/usr/local/src/sage-config/src/sage/structure/parent.pyx in sage.structure.parent.Parent.coerce (build/cythonized/sage/structure/parent.c:10829)()
   1157                 except Exception:
   1158                     _record_exception()
-> 1159             raise TypeError("no canonical coercion from %s to %s" % (parent(x), self))
   1160         else:
   1161             return (<map.Map>mor)._call_(x)

TypeError: no canonical coercion from Real Field with 53 bits of precision to Rational Field

CC: @videlec

Component: algebra

Author: Jeroen Demeyer

Branch/Commit: f65afd0

Reviewer: Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/5574

[burcin]

comment:1

From sage-support @freenode today:

<base3___> raising a matrix to a variable power fails with 
NotImplementedError: non-integral exponents not supported,
even if i do assume(n, 'integer')
<base3___> am i missing something or should i file this as a bug?
<burcin> base3___, it's a bug

[rwst]

Description changed:

--- 
+++ 
@@ -1,4 +1,4 @@
-Reported by Alex Raichev on sage-support:
+The first case reported by Alex Raichev on sage-support:
 
 ```
 sage: var('n',ns=1)
@@ -24,6 +24,14 @@
 
 TypeError: no canonical coercion from New Symbolic Ring to Rational
 Field
+
+also:
+
+sage: (QQbar(2)^3)^1.
+...
+TypeError: no canonical coercion from Real Field with 53 bits of precision to Rational Field
 ```
 
-Since pynac supports using arbitrary Sage objects as numeric objects in symbolic expressions, we should return a symbolic expression as a result of the above commands. 
+case 1: Since pynac supports using arbitrary Sage objects as numeric objects in symbolic expressions, we should return a symbolic expression as a result of the above commands. 
+
+case 2: just coerce to CDF

[rwst]

comment:6

This is a general shortcoming of AlgebraicNumber.__pow__():

sage: (QQbar(2)^3)^1.
...
TypeError: no canonical coercion from Real Field with 53 bits of precision to Rational Field