Trac #20756: sign is slow (if not wrong) for number field elements

{{{
sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^100 - 3, embedding=AA(3)**(1/100))
sage: %time sign(a)
CPU times: user 1.34 s, sys: 0 ns, total: 1.34 s
Wall time: 1.33 s
1
sage: b = continued_fraction(a).convergent(20) - a
sage: sign(b)
sgn(-1/1495877943276*(1/4)^(1/5)*(-(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) +
1)*(-3^(1/4))^(1/5))^(1/5)*(373969485819*sqrt(5) -
373969485819*I*sqrt(-2*sqrt(5) + 10) + 373969485819) +
378100611523/373969485819)
}}}
With the branch applied
{{{
sage: %time sign(a)
CPU times: user 4 ms, sys: 0 ns, total: 4 ms
Wall time: 4.65 ms
1
sage: b = continued_fraction(a).convergent(20) - a
sage: %time sign(b)
CPU times: user 4 ms, sys: 0 ns, total: 4 ms
Wall time: 6.02 ms
-1
}}}

URL: http://trac.sagemath.org/20756
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Marc Mezzarobba

src/sage/rings/number_field/number_field_element.pyx

@@ -930,6 +930,58 @@ cdef class NumberFieldElement(FieldElement):
         """
         return self.abs(prec=53, i=None)
 
+    def sign(self):
+        r"""
+        Return the sign of this algebraic number (if a real embedding is well
+        defined)
+
+        EXAMPLES::
+
+
+            sage: K.<a> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))
+            sage: K.zero().sign()
+            0
+            sage: K.one().sign()
+            1
+            sage: (-K.one()).sign()
+            -1
+            sage: a.sign()
+            1
+            sage: (a - 234917380309015/186454048314072).sign()
+            1
+            sage: (a - 3741049304830488/2969272800976409).sign()
+            -1
+
+        If the field is not embedded in real numbers, this method will only work
+        for rational elements::
+
+            sage: L.<b> = NumberField(x^4 - x - 1)
+            sage: b.sign()
+            Traceback (most recent call last):
+            ...
+            TypeError: sign not well defined since no real embedding is
+            specified
+            sage: L(-33/125).sign()
+            -1
+            sage: L.zero().sign()
+            0
+        """
+        if ZZX_deg(self.__numerator) == -1:
+            return 0
+        if ZZX_deg(self.__numerator) == 0:
+            return ZZ_sign(ZZX_coeff(self.__numerator, 0))
+
+        if not (<number_field_base.NumberField> self._parent)._embedded_real:
+            raise TypeError("sign not well defined since no real embedding is specified")
+
+        from sage.rings.real_mpfi import RealIntervalField
+        i = 0
+        a = RealIntervalField(53)(self)
+        while a.contains_zero():
+            i += 1
+            a = RealIntervalField(53<<i)(self)
+        return a.unique_sign()
+
     def floor(self):
         r"""
         Return the floor of this number field element.
@@ -968,7 +1020,7 @@ cdef class NumberFieldElement(FieldElement):
             ...
             TypeError: floor not uniquely defined since no real embedding is specified
         """
-        if ZZX_deg(self.__numerator) == 0:
+        if ZZX_deg(self.__numerator) <= 0:
             return self._rational_().floor()
 
         if not (<number_field_base.NumberField> self._parent)._embedded_real:
@@ -1024,7 +1076,7 @@ cdef class NumberFieldElement(FieldElement):
             ...
             TypeError: ceil not uniquely defined since no real embedding is specified
         """
-        if ZZX_deg(self.__numerator) == 0:
+        if ZZX_deg(self.__numerator) <= 0:
             return self._rational_().ceil()
 
         if not (<number_field_base.NumberField> self._parent)._embedded_real: