#18282 Improve pol(expr) for expr in SR

In particular, it should not return an expression in Horner form.

src/sage/calculus/calculus.py

@@ -889,7 +889,7 @@ def minpoly(ex, var='x', algorithm=None, bits=None, degree=None, epsilon=0):
         sage: f = a.minpoly(); f
         x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576
         sage: f(a)
-        ((((sqrt(5) + sqrt(3) + sqrt(2))^2 - 40)*(sqrt(5) + sqrt(3) + sqrt(2))^2 + 352)*(sqrt(5) + sqrt(3) + sqrt(2))^2 - 960)*(sqrt(5) + sqrt(3) + sqrt(2))^2 + 576
+        (sqrt(5) + sqrt(3) + sqrt(2))^8 - 40*(sqrt(5) + sqrt(3) + sqrt(2))^6 + 352*(sqrt(5) + sqrt(3) + sqrt(2))^4 - 960*(sqrt(5) + sqrt(3) + sqrt(2))^2 + 576
         sage: f(a).expand()
         0
 
@@ -906,7 +906,7 @@ def minpoly(ex, var='x', algorithm=None, bits=None, degree=None, epsilon=0):
         NotImplementedError: Could not prove minimal polynomial x^4 - 5/4*x^2 + 5/16 (epsilon 0.00000000000000e-1)
         sage: f = a.minpoly(algorithm='numerical', epsilon=1e-100); f
         x^4 - 5/4*x^2 + 5/16
-        sage: f(a).numerical_approx(100)
+        sage: f(a).horner(a).numerical_approx(100)
         0.00000000000000000000000000000
 
     The degree must be high enough (default tops out at 24).

src/sage/functions/orthogonal_polys.py

@@ -1225,7 +1225,7 @@ def gen_legendre_P(n,m,x):
         sage: gen_legendre_P(3, 1, t)
         -3/2*(5*t^2 - 1)*sqrt(-t^2 + 1)
         sage: gen_legendre_P(4, 3, t)
-        105*(t^2 - 1)*sqrt(-t^2 + 1)*t
+        105*(t^3 - t)*sqrt(-t^2 + 1)
         sage: gen_legendre_P(7, 3, I).expand()
         -16695*sqrt(2)
         sage: gen_legendre_P(4, 1, 2.5)

src/sage/rings/number_field/number_field_element.pyx

@@ -2293,7 +2293,7 @@ cdef class NumberFieldElement(FieldElement):
             sage: SR(b.minpoly()).solve(SR('x'), explicit_solutions=True)
             []
             sage: SR(b)
-            1/8*((sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) - 4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) + 5)^2 + 4)*(sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) - 4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) + 5)
+            1/8*(sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) - 4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) + 5)^3 + 1/2*sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) - 4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) + 5/2
 
         """
         K = self._parent.fraction_field()

src/sage/rings/polynomial/polynomial_element.pyx

@@ -715,6 +715,15 @@ cdef class Polynomial(CommutativeAlgebraElement):
                     return self.reverse()(a_inverse) / a_inverse**self.degree()
             except AttributeError:
                 pass
+        from sage.symbolic.ring import SR
+        if parent(a) is SR:
+            if len(x) == 1:
+                try:
+                    return SR(self(a.pyobject()))
+                except TypeError:
+                    return SR.zero().add(*(self[i]*a**i for i in xrange(d + 1)))
+            else:
+                return SR.zero().add(*(self[i](other_args)*a**i for i in xrange(d + 1)))
 
         i = d - 1
         if len(x) > 1:
@@ -1072,7 +1081,7 @@ cdef class Polynomial(CommutativeAlgebraElement):
             sage: R.<x> = QQ[]
             sage: f = x^3 + x
             sage: g = f._symbolic_(SR); g
-            (x^2 + 1)*x
+            x^3 + x
             sage: g(x=2)
             10
 

src/sage/rings/polynomial/polynomial_zz_pex.pyx

@@ -251,7 +251,7 @@ cdef class Polynomial_ZZ_pEX(Polynomial_template):
             sage: P.<y> = F[]
             sage: p = y^4 + x*y^3 + y^2 + (x + 1)*y + x + 1
             sage: SR(p)      #indirect doctest
-            (((y + x)*y + 1)*y + x + 1)*y + x + 1
+            y^4 + x*y^3 + y^2 + (x + 1)*y + x + 1
             sage: p(2)
             x + 1
             sage: p(y=2)