trac 2693 resultants for polynomials over inexact rings

Author: fchapoton

src/sage/rings/polynomial/multi_polynomial_element.py

@@ -50,7 +50,7 @@
 #
 #  The full text of the GPL is available at:
 #
-#                  http://www.gnu.org/licenses/
+#                  https://www.gnu.org/licenses/
 #*****************************************************************************
 from __future__ import absolute_import
 from six.moves import range
@@ -1880,6 +1880,9 @@ def resultant(self, other, variable=None):
         If a second argument is not provided, the first variable of
         ``self.parent()`` is chosen.
 
+        For inexact rings or rings not available in Singular,
+        this computes the determinant of the Sylvester matrix.
+
         INPUT:
 
         - ``other`` -- polynomial in ``self.parent()``
@@ -1914,11 +1917,18 @@ def resultant(self, other, variable=None):
             sage: (x^2 + 1).resultant(x^2 - y)
             y^2 + 2*y + 1
 
+        Test for :trac:`2693`::
+
+            sage: R.<x,y> = RR[]
+            sage: p = x + y
+            sage: q = x*y
+            sage: p.resultant(q)
+            -y^2
         """
         R = self.parent()
         if variable is None:
             variable = R.gen(0)
-        if R._has_singular:
+        if R._has_singular and R.is_exact():
             rt = self._singular_().resultant(other._singular_(), variable._singular_())
             r = rt.sage_poly(R)
         else:

src/sage/rings/polynomial/polynomial_element.pyx

@@ -6306,13 +6306,20 @@ cdef class Polynomial(CommutativeAlgebraElement):
 
             sage: A.<a,b,c> = Frac(PolynomialRing(QQ,'a,b,c'))
             sage: B.<d,e,f> = PolynomialRing(A,'d,e,f')
-            sage: R.<x>= PolynomialRing(B,'x')
+            sage: R.<x> = PolynomialRing(B,'x')
             sage: S.<y> = PolynomialRing(R,'y')
             sage: p = ((1/b^2*d^2+1/a)*x*y^2+a*b/c*y+e+x^2)
             sage: q = -4*c^2*y^3+1
             sage: p.resultant(q)
             16*c^4*x^6 + 48*c^4*e*x^4 + (1/b^6*d^6 + 3/(a*b^4)*d^4 + ((-12*a^3*b*c + 3)/(a^2*b^2))*d^2 + (-12*a^3*b*c + 1)/a^3)*x^3 + 48*c^4*e^2*x^2 + (((-12*a*c)/b)*d^2*e + (-12*b*c)*e)*x + 16*c^4*e^3 + 4*a^3*b^3/c
 
+        Test for :trac:`10978`::
+
+            sage: R.<x> = PolynomialRing(CDF)
+            sage: f = R(1 - I*x + (0.5)*x^2 + (1.7)*x^3)
+            sage: g = f.derivative()
+            sage: f.resultant(g)
+            133.92599999999996 + 37.56999999999999*I
         """
         variable = self.variable_name()
         try: