Trac #23835: Replace Maxima with Pynac/Singular in Expression.factor()

The default for symbolic factorization should be changed. Maxima as
default should be replaced with a call to Pynac's `factor()`
implementation, which itself uses Singular at the moment. Maxima should
be made available via `ex.factor(algorithm=...)`.

As an example of performance gain here one of the Fateman benchmarks:
{{{
sage: var('a b c k s y z')
(a, b, c, k, s, y, z)
sage: f = (1+x+y+z)^20+1
sage: g = (f*(f+1)).expand()
sage: %time _=g.factor()
}}}
This takes 11 seconds on 8.1.beta5 and 3.3 seconds with Pynac factor
(identical time using polynomial ring).

URL: https://trac.sagemath.org/23835
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Travis Scrimshaw

src/sage/coding/code_bounds.py

@@ -580,7 +580,7 @@ def entropy(x, q=2):
         sage: codes.bounds.entropy(0, 2)
         0
         sage: codes.bounds.entropy(1/5,4).factor()
-        1/10*(log(5) + log(3) - 4*log(4/5))/log(2)
+        1/10*(log(3) - 4*log(4/5) - log(1/5))/log(2)
         sage: codes.bounds.entropy(1, 3)
         log(2)/log(3)
 

src/sage/symbolic/expression.pyx

@@ -4116,7 +4116,7 @@ cdef class Expression(CommutativeRingElement):
             sage: g = derivative(f, x); g # this is a complex expression
             -1/2*((x^2 + 1)*x/(x^2 - 1)^2 - x/(x^2 - 1))/((x^2 + 1)/(x^2 - 1))^(3/4)
             sage: g.factor()
-            -x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/(x^2 - 1))^(3/4))
+            -x/((x + 1)^2*(x - 1)^2*((x^2 + 1)/((x + 1)*(x - 1)))^(3/4))
 
         ::
 
@@ -11141,22 +11141,24 @@ cdef class Expression(CommutativeRingElement):
             sage: (f(x).diff(x)^2-1).factor()
             (diff(f(x), x) + 1)*(diff(f(x), x) - 1)
         """
-        from sage.calculus.calculus import symbolic_expression_from_maxima_string, symbolic_expression_from_string
+        from sage.calculus.calculus import symbolic_expression_from_maxima_string
+        cdef GEx x
+        cdef bint b
         if dontfactor:
             m = self._maxima_()
             name = m.name()
             varstr = ','.join(['_SAGE_VAR_' + str(v) for v in dontfactor])
             cmd = 'block([dontfactor:[%s]],factor(%s))' % (varstr, name)
             return symbolic_expression_from_maxima_string(cmd)
+        sig_on()
+        try:
+            b = g_factor(self._gobj, x)
+        finally:
+            sig_off()
+        if b:
+            return new_Expression_from_GEx(self._parent, x)
         else:
-            try:
-                from sage.rings.all import QQ
-                f = self.polynomial(QQ)
-                w = repr(f.factor())
-                return symbolic_expression_from_string(w)
-            except (TypeError, NotImplementedError):
-                pass
-            return self.parent()(self._maxima_().factor())
+            return self
 
     def factor_list(self, dontfactor=[]):
         """