Trac #23808: Keep embedding info when converting algebraics to number…

… field in projective morphism

Given a map or point defined over QQbar, you can call
_numberfield_from_algebraics to return a map over a number field.
However, the resulting object cannot be converted back to QQbar since
the embedding information is lost.

URL: https://trac.sagemath.org/23808
Reported by: paulfili
Ticket author(s): Paul Fili
Reviewer(s): David Roe, Ben Hutz

src/sage/schemes/projective/projective_morphism.py

@@ -1295,7 +1295,7 @@ def degree_sequence(self, iterates=2):
 
             sage: P2.<X,Y,Z> = ProjectiveSpace(QQ, 2)
             sage: H = End(P2)
-            sage: f = H([Z^2, X*Y, Y^2]) 
+            sage: f = H([Z^2, X*Y, Y^2])
             sage: f.degree_sequence(15)
             [2, 3, 5, 8, 11, 17, 24, 31, 45, 56, 68, 91, 93, 184, 275]
 
@@ -1304,23 +1304,23 @@ def degree_sequence(self, iterates=2):
             sage: F.<t> = PolynomialRing(QQ)
             sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2)
             sage: H = End(P2)
-            sage: f = H([Y*Z, X*Y, Y^2 + t*X*Z]) 
+            sage: f = H([Y*Z, X*Y, Y^2 + t*X*Z])
             sage: f.degree_sequence(5)
             [2, 3, 5, 8, 13]
 
         ::
 
             sage: P2.<X,Y,Z> = ProjectiveSpace(QQ, 2)
             sage: H = End(P2)
-            sage: f = H([X^2, Y^2, Z^2]) 
+            sage: f = H([X^2, Y^2, Z^2])
             sage: f.degree_sequence(10)
             [2, 4, 8, 16, 32, 64, 128, 256, 512, 1024]
 
         ::
 
             sage: P2.<X,Y,Z> = ProjectiveSpace(ZZ, 2)
             sage: H = End(P2)
-            sage: f = H([X*Y, Y*Z+Z^2, Z^2]) 
+            sage: f = H([X*Y, Y*Z+Z^2, Z^2])
             sage: f.degree_sequence(10)
             [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
         """
@@ -5196,12 +5196,40 @@ def _number_field_from_algebraics(self):
               To:   Projective Space of dimension 2 over Number Field in a with defining polynomial y^4 + 3*y^2 + 1
               Defn: Defined on coordinates by sending (z0 : z1) to
                     (z0^2 + (a^3 + 2*a)*z0*z1 + 3*z1^2 : z1^2 : (2*a^2 + 3)*z0*z1)
+
+        The following was fixed in :trac:`23808`::
+
+            sage: R.<t>=PolynomialRing(QQ)
+            sage: s = (t^3+t+1).roots(QQbar)[0][0]
+            sage: P.<x,y>=ProjectiveSpace(QQbar,1)
+            sage: H = Hom(P,P)
+            sage: f = H([s*x^3-13*y^3, y^3-15*y^3])
+            sage: f
+            Scheme endomorphism of Projective Space of dimension 1 over Algebraic Field
+              Defn: Defined on coordinates by sending (x : y) to
+                    ((-0.6823278038280193?)*x^3 + (-13)*y^3 : (-14)*y^3)
+            sage: f_alg = f._number_field_from_algebraics()
+            sage: f_alg.change_ring(QQbar) # Used to fail
+            Scheme endomorphism of Projective Space of dimension 1 over Algebraic Field
+              Defn: Defined on coordinates by sending (z0 : z1) to
+                    ((-0.6823278038280193?)*z0^3 + (-13)*z1^3 : (-14)*z1^3)
+
         """
         from sage.schemes.projective.projective_space import is_ProjectiveSpace
         if not (is_ProjectiveSpace(self.domain()) and is_ProjectiveSpace(self.domain())):
             raise NotImplementedError("not implemented for subschemes")
 
-        K,C,phi = number_field_elements_from_algebraics([c for f in self for c in f.coefficients()])
+        K_pre,C,phi = number_field_elements_from_algebraics([c for f in self for c in f.coefficients()])
+        # Trac 23808: The field K_pre returned above does not have its embedding set to be phi
+        # and phi is forgotten, so we redefine K_pre to be a field K with phi as the specified
+        # embedding:
+        if K_pre is QQ:
+            K = QQ
+        else:
+            from sage.rings.number_field.number_field import NumberField
+            K = NumberField(K_pre.polynomial(), embedding=phi(K_pre.gen()), name='a')
+            psi = K_pre.hom([K.gen()], K) # Identification of K_pre with K
+            C = [ psi(c) for c in C ] # The elements of C were in K_pre, move them to K
         from sage.schemes.projective.projective_space import ProjectiveSpace
         N = self.domain().dimension_relative()
         PS = ProjectiveSpace(K,N,'z')

src/sage/schemes/projective/projective_point.py

@@ -1748,12 +1748,36 @@ def _number_field_from_algebraics(self):
             (1/2*a^3 + a^2 - 1/2*a : 1)
             sage: S.codomain()
             Projective Space of dimension 1 over Number Field in a with defining polynomial y^4 + 1
+
+        The following was fixed in :trac:`23808`::
+
+            sage: R.<x> = PolynomialRing(QQ)
+            sage: P.<x,y> = ProjectiveSpace(QQbar,1)
+            sage: Q = P([-1/2*QQbar(sqrt(2)) + QQbar(I), 1]);Q
+            (-0.7071067811865475? + 1*I : 1)
+            sage: S = Q._number_field_from_algebraics(); S
+            (1/2*a^3 + a^2 - 1/2*a : 1)
+            sage: T = S.change_ring(QQbar) # Used to fail
+            sage: T
+            (-0.7071067811865475? + 1.000000000000000?*I : 1)
+            sage: Q[0] == T[0]
+            True
         """
         from sage.schemes.projective.projective_space import is_ProjectiveSpace
         if not is_ProjectiveSpace(self.codomain()):
             raise NotImplementedError("not implemented for subschemes")
 
-        K,P,phi = number_field_elements_from_algebraics(list(self))
+        # Trac #23808: Keep the embedding info associated with the number field K
+        # used below, instead of in the separate embedding map phi which is
+        # forgotten.
+        K_pre,P,phi = number_field_elements_from_algebraics(list(self))
+        if K_pre is QQ:
+            K = QQ
+        else:
+            from sage.rings.number_field.number_field import NumberField
+            K = NumberField(K_pre.polynomial(), embedding=phi(K_pre.gen()), name='a')
+            psi = K_pre.hom([K.gen()], K) # Identification of K_pre with K
+            P = [ psi(p) for p in P ] # The elements of P were elements of K_pre
         from sage.schemes.projective.projective_space import ProjectiveSpace
         PS = ProjectiveSpace(K,self.codomain().dimension_relative(),'z')
         return(PS(P))