19552: forward and preimages of projective subschemes

src/sage/schemes/generic/algebraic_scheme.py

@@ -130,7 +130,7 @@
 #          class AlgebraicScheme_subscheme_affine_toric
 #    class AlgebraicScheme_quasi
 
-
+from copy import copy
 from sage.categories.number_fields import NumberFields
 
 from sage.rings.all import ZZ
@@ -2208,6 +2208,112 @@ def is_smooth(self, point=None):
         self._smooth = (sing_dim <= 0)
         return self._smooth
 
+    def orbit(self, f, N):
+        r"""
+        Returns the orbit of `self` by ``f``. If `N` is an integer it returns `[self,f(self),\ldots,f^N(self)]`.
+        If `N` is a list or tuple `N=[m,k]` it returns `[f^m(self),\ldots,f^k(self)`].
+        Perform the checks on subscheme initialization if ``check=True``
+
+        INPUT:
+
+        - ``f`` -- a :class:`SchemeMorphism_polynomial` with ``self`` in ``f.domain()``
+
+        - ``N`` -- a non-negative integer or list or tuple of two non-negative integers
+
+        OUTPUT:
+
+        - a proejctive subscheme
+
+        EXAMPLES::
+
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: H = End(P)
+            sage: f = H([(x-2*y)^2,(x-2*z)^2,(x-2*w)^2,x^2])
+            sage: f.orbit(P.subscheme([x]),5)
+            [Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+               x,
+             Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+               w,
+             Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+               z - w,
+             Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+               y - z,
+             Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+               x - y,
+             Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+               x - w]
+        """
+        if not f.is_endomorphism():
+            raise TypeError("Map must be an endomorphism for iteration.")
+        if (isinstance(N,(list,tuple)) == False):
+            N = [0,N]
+        try:
+            N[0] = ZZ(N[0])
+            N[1] = ZZ(N[1])
+        except TypeError:
+            raise TypeError("Orbit bounds must be integers")
+        if N[0] < 0 or N[1] < 0:
+            raise TypeError("Orbit bounds must be non-negative")
+        if N[0] > N[1]:
+            return([])
+
+        Q = copy(self)
+        for i in range(1, N[0]+1):
+            Q = f(Q)
+        Orb = [Q]
+
+        for i in range(N[0]+1, N[1]+1):
+            Q = f(Q)
+            Orb.append(Q)
+        return(Orb)
+
+    def nth_iterate(self, f, n , **kwds):
+        r"""
+        For a subscheme ``self`` and a map `f` this function returns
+        the nth iterate of `self` by ``f``.
+
+        INPUT:
+
+        - ``f`` -- a SchmemMorphism_polynomial with ``self`` in ``f.domain()``
+
+        - ``n`` -- a positive integer.
+
+        OUTPUT:
+
+        - A subscheme in ``f.codomain()``
+
+        EXAMPLES::
+
+            sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
+            sage: H = End(P)
+            sage: f = H([y^2, z^2, x^2, w^2])
+            sage: f.nth_iterate(P.subscheme([x-w,y-z]), 3)
+            Closed subscheme of Projective Space of dimension 3 over Rational Field
+            defined by:
+              y - z,
+              x - w
+        """
+        if not f.is_endomorphism():
+            raise TypeError("Map must be an endomorphism for iteration.")
+        try:
+            n = ZZ(n)
+        except TypeError:
+            raise TypeError("Iterate number must be an integer")
+        if n < 0:
+            raise TypeError("Must be a forward orbit")
+        if n == 0:
+            return(self)
+        else:
+            Q = f(self)
+            for i in range(2,n+1):
+                Q = f(Q)
+            return(Q)
 
 class AlgebraicScheme_subscheme_product_projective(AlgebraicScheme_subscheme_projective):
 

src/sage/schemes/generic/morphism.py

@@ -1050,12 +1050,7 @@ def _call_(self, x):
             sage: H=Hom(X,X)
             sage: f=H([x^2,y^2,z^2]);
             sage: f(P([4,4,1]))
-            Traceback (most recent call last):
-            ...
-            TypeError: (4 : 4 : 1) fails to convert into the map's domain
-            Closed subscheme of Projective Space of dimension 2 over Integer
-            Ring defined by:
-              x^2 - y^2, but a `pushforward` method is not properly implemented
+            (16 : 16 : 1)
         """
         # Checks were done in __call__
         P = [f(x._coords) for f in self.defining_polynomials()]
@@ -1134,16 +1129,16 @@ def _call_with_args(self, x, args, kwds):
 
         ::
 
-            sage: P.<x,y,z>=ProjectiveSpace(ZZ,2)
-            sage: X=P.subscheme(x^2-y^2);
-            sage: H=Hom(X,X)
-            sage: f=H([x^2,y^2,z^2]);
-            sage: f(P([4,4,1]))
+            sage: P.<x,y,z> = ProjectiveSpace(ZZ, 2)
+            sage: P2.<u,v,w,t> = ProjectiveSpace(ZZ, 3)
+            sage: X = P.subscheme(x^2-y^2);
+            sage: H = Hom(X, X)
+            sage: f = H([x^2, y^2, z^2]);
+            sage: f(P2([4,4,1,1]))
             Traceback (most recent call last):
             ...
-            TypeError: (4 : 4 : 1) fails to convert into the map's domain
-            Closed subscheme of Projective Space of dimension 2 over
-            Integer Ring defined by:
+            TypeError: (4 : 4 : 1 : 1) fails to convert into the map's domain Closed subscheme of
+            Projective Space of dimension 2 over Integer Ring defined by:
               x^2 - y^2, but a `pushforward` method is not properly implemented
         """
         if args: