Scheme morphism for points on affine varieties
AUTHORS:
Bases: sage.schemes.generic.morphism.SchemeMorphism_point
A rational point on an affine scheme.
INPUT:
EXAMPLES:
sage: A = AffineSpace(2, QQ)
sage: A(1,2)
(1, 2)
Returns the point
INPUT:
OUTPUT:
EXAMPLES:
sage: A.<x,y>=AffineSpace(QQ,2)
sage: H=Hom(A,A)
sage: f=H([(x-2*y^2)/x,3*x*y])
sage: A(9,3).nth_iterate(f,3)
(-104975/13123, -9566667)
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: X=A.subscheme([x-y^2])
sage: H=Hom(X,X)
sage: f=H([9*y^2,3*y])
sage: X(9,3).nth_iterate(f,4)
(59049, 243)
Returns the orbit of self by . If
is an integer it returns
.
If
is a list or tuple
it returns
.
INPUT:
OUTPUT:
EXAMPLES:
sage: A.<x,y>=AffineSpace(QQ,2)
sage: H=Hom(A,A)
sage: f=H([(x-2*y^2)/x,3*x*y])
sage: A(9,3).orbit(f,3)
[(9, 3), (-1, 81), (13123, -243), (-104975/13123, -9566667)]
sage: A.<x>=AffineSpace(QQ,1)
sage: H=Hom(A,A)
sage: f=H([(x-2)/x])
sage: A(1/2).orbit(f,[1,3])
[(-3), (5/3), (-1/5)]
sage: A.<x,y>=AffineSpace(ZZ,2)
sage: X=A.subscheme([x-y^2])
sage: H=Hom(X,X)
sage: f=H([9*y^2,3*y])
sage: X(9,3).orbit(f,(0,4))
[(9, 3), (81, 9), (729, 27), (6561, 81), (59049, 243)]
Bases: sage.schemes.affine.affine_point.SchemeMorphism_point_affine
The Python constructor.
See SchemeMorphism_point_affine for details.
TESTS:
sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
sage: A3.<x,y,z> = AffineSpace(QQ, 3)
sage: SchemeMorphism_point_affine(A3(QQ), [1,2,3])
(1, 2, 3)
Bases: sage.schemes.affine.affine_point.SchemeMorphism_point_affine_field
The Python constructor.
See SchemeMorphism_point_affine for details.
TESTS:
sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
sage: A3.<x,y,z> = AffineSpace(QQ, 3)
sage: SchemeMorphism_point_affine(A3(QQ), [1,2,3])
(1, 2, 3)
Every points is preperiodic over a finite field. This funtion returns the pair where
is the
preperiod and
the period of the point self by f.
INPUT:
OUTPUT:
EXAMPLES:
sage: P.<x,y,z>=AffineSpace(GF(5),3)
sage: H=Hom(P,P)
sage: f=H([x^2+y^2,y^2,z^2+y*z])
sage: P(1,1,1).orbit_structure(f)
[0, 6]
sage: P.<x,y,z>=AffineSpace(GF(7),3)
sage: X=P.subscheme(x^2-y^2)
sage: H=Hom(X,X)
sage: f=H([x^2,y^2,z^2])
sage: X(1,1,2).orbit_structure(f)
[0, 2]
sage: P.<x,y>=AffineSpace(GF(13),2)
sage: H=Hom(P,P)
sage: f=H([x^2-y^2,y^2])
sage: P(3,4).orbit_structure(f)
[2, 6]