Generic dynamical systems on schemes

This is the generic class for dynamical systems and contains the exported constructor functions. The constructor functions can take either polynomials (or rational functions in the affine case) or morphisms from which to construct a dynamical system. If the domain is not specified, it is constructed. However, if you plan on working with points or subvarieties in the domain, it recommended to specify the domain. For products of projective spaces the domain must be specified.

The initialization checks are always performed by the constructor functions. It is possible, but not recommended, to skip these checks by calling the class initialization directly.

AUTHORS:

  • Ben Hutz (July 2017): initial version

class sage.dynamics.arithmetic_dynamics.generic_ds.DynamicalSystem(polys_or_rat_fncts, domain)[source]

Bases: SchemeMorphism_polynomial

Base class for dynamical systems of schemes.

INPUT:

  • polys_or_rat_fncts – list of polynomials or rational functions, all of which should have the same parent

  • domain – an affine or projective scheme, or product of projective schemes, on which polys defines an endomorphism (Subschemes are also ok)

  • names – (default: ('X', 'Y')) tuple of strings to be used as coordinate names for a projective space that is constructed

    The following combinations of morphism_or_polys and domain are meaningful:

    • morphism_or_polys is a SchemeMorphism; domain is ignored in this case

    • morphism_or_polys is a list of homogeneous polynomials that define a rational endomorphism of domain

    • morphism_or_polys is a list of homogeneous polynomials and domain is unspecified; domain is then taken to be the projective space of appropriate dimension over the common parent of the elements in morphism_or_polys

    • morphism_or_polys is a single polynomial or rational function; domain is ignored and taken to be a 1-dimensional projective space over the base ring of morphism_or_polys with coordinate names given by names

EXAMPLES:

sage: A.<x> = AffineSpace(QQ,1)
sage: f = DynamicalSystem_affine([x^2 + 1])
sage: type(f)
<class 'sage.dynamics.arithmetic_dynamics.affine_ds.DynamicalSystem_affine_field'>

>>> from sage.all import *
>>> A = AffineSpace(QQ,Integer(1), names=('x',)); (x,) = A._first_ngens(1)
>>> f = DynamicalSystem_affine([x**Integer(2) + Integer(1)])
>>> type(f)
<class 'sage.dynamics.arithmetic_dynamics.affine_ds.DynamicalSystem_affine_field'>

A.<x> = AffineSpace(QQ,1)
f = DynamicalSystem_affine([x^2 + 1])
type(f)
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: f = DynamicalSystem_projective([x^2 + y^2, y^2])
sage: type(f)
<class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective_field'>

>>> from sage.all import *
>>> P = ProjectiveSpace(QQ,Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2) + y**Integer(2), y**Integer(2)])
>>> type(f)
<class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective_field'>

P.<x,y> = ProjectiveSpace(QQ,1)
f = DynamicalSystem_projective([x^2 + y^2, y^2])
type(f)
>>> from sage.all import *
>>> P = ProjectiveSpace(QQ,Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2) + y**Integer(2), y**Integer(2)])
>>> type(f)
<class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective_field'>

P.<x,y> = ProjectiveSpace(QQ,1)
f = DynamicalSystem_projective([x^2 + y^2, y^2])
type(f)
sage: P1.<x,y> = ProjectiveSpace(CC,1)
sage: H = End(P1)
sage: DynamicalSystem(H([y, x]))
Dynamical System of Projective Space of dimension 1 over Complex Field
with 53 bits of precision
  Defn: Defined on coordinates by sending (x : y) to (y : x)

>>> from sage.all import *
>>> P1 = ProjectiveSpace(CC,Integer(1), names=('x', 'y',)); (x, y,) = P1._first_ngens(2)
>>> H = End(P1)
>>> DynamicalSystem(H([y, x]))
Dynamical System of Projective Space of dimension 1 over Complex Field
with 53 bits of precision
  Defn: Defined on coordinates by sending (x : y) to (y : x)

P1.<x,y> = ProjectiveSpace(CC,1)
H = End(P1)
DynamicalSystem(H([y, x]))
>>> from sage.all import *
>>> P1 = ProjectiveSpace(CC,Integer(1), names=('x', 'y',)); (x, y,) = P1._first_ngens(2)
>>> H = End(P1)
>>> DynamicalSystem(H([y, x]))
Dynamical System of Projective Space of dimension 1 over Complex Field
with 53 bits of precision
  Defn: Defined on coordinates by sending (x : y) to (y : x)

P1.<x,y> = ProjectiveSpace(CC,1)
H = End(P1)
DynamicalSystem(H([y, x]))

DynamicalSystem defaults to projective:

sage: R.<x,y,z> = QQ[]
sage: DynamicalSystem([x^2, y^2, z^2])
Dynamical System of Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x : y : z) to
        (x^2 : y^2 : z^2)

>>> from sage.all import *
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> DynamicalSystem([x**Integer(2), y**Integer(2), z**Integer(2)])
Dynamical System of Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x : y : z) to
        (x^2 : y^2 : z^2)

R.<x,y,z> = QQ[]
DynamicalSystem([x^2, y^2, z^2])
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: DynamicalSystem([y, x], domain=A)
Dynamical System of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to (y, x)
sage: H = End(A)
sage: DynamicalSystem(H([y, x]))
Dynamical System of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to (y, x)

>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> DynamicalSystem([y, x], domain=A)
Dynamical System of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to (y, x)
>>> H = End(A)
>>> DynamicalSystem(H([y, x]))
Dynamical System of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to (y, x)

A.<x,y> = AffineSpace(QQ, 2)
DynamicalSystem([y, x], domain=A)
H = End(A)
DynamicalSystem(H([y, x]))
>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> DynamicalSystem([y, x], domain=A)
Dynamical System of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to (y, x)
>>> H = End(A)
>>> DynamicalSystem(H([y, x]))
Dynamical System of Affine Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x, y) to (y, x)

A.<x,y> = AffineSpace(QQ, 2)
DynamicalSystem([y, x], domain=A)
H = End(A)
DynamicalSystem(H([y, x]))

Note that domain is ignored if an endomorphism is passed in:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P2.<x,y> = ProjectiveSpace(CC, 1)
sage: H = End(P2)
sage: f = H([CC.0*x^2, y^2])
sage: g = DynamicalSystem(f, domain=P)
sage: g.domain()
Projective Space of dimension 1 over Complex Field with 53 bits of precision

>>> from sage.all import *
>>> P = ProjectiveSpace(QQ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> P2 = ProjectiveSpace(CC, Integer(1), names=('x', 'y',)); (x, y,) = P2._first_ngens(2)
>>> H = End(P2)
>>> f = H([CC.gen(0)*x**Integer(2), y**Integer(2)])
>>> g = DynamicalSystem(f, domain=P)
>>> g.domain()
Projective Space of dimension 1 over Complex Field with 53 bits of precision

P.<x,y> = ProjectiveSpace(QQ, 1)
P2.<x,y> = ProjectiveSpace(CC, 1)
H = End(P2)
f = H([CC.0*x^2, y^2])
g = DynamicalSystem(f, domain=P)
g.domain()

Constructing a common parent:

sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
sage: DynamicalSystem([CC.0*x^2, 4/5*y^2])
Dynamical System of
 Projective Space of dimension 1 over Complex Field with 53 bits of precision
  Defn: Defined on coordinates by sending (x : y) to
        (1.00000000000000*I*x^2 : 0.800000000000000*y^2)

sage: # needs sage.rings.finite_rings
sage: P.<x,y> = ProjectiveSpace(GF(5), 1)
sage: K.<t> = GF(25)
sage: DynamicalSystem([GF(5)(3)*x^2, K(t)*y^2])
Dynamical System of Projective Space of dimension 1 over Finite Field in t of size 5^2
  Defn: Defined on coordinates by sending (x : y) to
        (-2*x^2 : t*y^2)

>>> from sage.all import *
>>> P = ProjectiveSpace(ZZ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> DynamicalSystem([CC.gen(0)*x**Integer(2), Integer(4)/Integer(5)*y**Integer(2)])
Dynamical System of
 Projective Space of dimension 1 over Complex Field with 53 bits of precision
  Defn: Defined on coordinates by sending (x : y) to
        (1.00000000000000*I*x^2 : 0.800000000000000*y^2)

>>> # needs sage.rings.finite_rings
>>> P = ProjectiveSpace(GF(Integer(5)), Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> K = GF(Integer(25), names=('t',)); (t,) = K._first_ngens(1)
>>> DynamicalSystem([GF(Integer(5))(Integer(3))*x**Integer(2), K(t)*y**Integer(2)])
Dynamical System of Projective Space of dimension 1 over Finite Field in t of size 5^2
  Defn: Defined on coordinates by sending (x : y) to
        (-2*x^2 : t*y^2)

P.<x,y> = ProjectiveSpace(ZZ, 1)
DynamicalSystem([CC.0*x^2, 4/5*y^2])
# needs sage.rings.finite_rings
P.<x,y> = ProjectiveSpace(GF(5), 1)
K.<t> = GF(25)
DynamicalSystem([GF(5)(3)*x^2, K(t)*y^2])
as_scheme_morphism()[source]

Return this dynamical system as SchemeMorphism_polynomial.

OUTPUT: SchemeMorphism_polynomial

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: f = DynamicalSystem_projective([x^2, y^2, z^2])
sage: type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space'>

>>> from sage.all import *
>>> P = ProjectiveSpace(ZZ, Integer(2), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> f = DynamicalSystem_projective([x**Integer(2), y**Integer(2), z**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space'>

P.<x,y,z> = ProjectiveSpace(ZZ, 2)
f = DynamicalSystem_projective([x^2, y^2, z^2])
type(f.as_scheme_morphism())
sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem_projective([x^2 - y^2, y^2])
sage: type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_field'>

>>> from sage.all import *
>>> P = ProjectiveSpace(QQ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2) - y**Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_field'>

P.<x,y> = ProjectiveSpace(QQ, 1)
f = DynamicalSystem_projective([x^2 - y^2, y^2])
type(f.as_scheme_morphism())
>>> from sage.all import *
>>> P = ProjectiveSpace(QQ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2) - y**Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_field'>

P.<x,y> = ProjectiveSpace(QQ, 1)
f = DynamicalSystem_projective([x^2 - y^2, y^2])
type(f.as_scheme_morphism())
sage: P.<x,y> = ProjectiveSpace(GF(5), 1)
sage: f = DynamicalSystem_projective([x^2, y^2])
sage: type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field'>

>>> from sage.all import *
>>> P = ProjectiveSpace(GF(Integer(5)), Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field'>

P.<x,y> = ProjectiveSpace(GF(5), 1)
f = DynamicalSystem_projective([x^2, y^2])
type(f.as_scheme_morphism())
>>> from sage.all import *
>>> P = ProjectiveSpace(GF(Integer(5)), Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field'>

P.<x,y> = ProjectiveSpace(GF(5), 1)
f = DynamicalSystem_projective([x^2, y^2])
type(f.as_scheme_morphism())
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: f = DynamicalSystem_affine([x^2 - 2, y^2])
sage: type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space'>

>>> from sage.all import *
>>> A = AffineSpace(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> f = DynamicalSystem_affine([x**Integer(2) - Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space'>

A.<x,y> = AffineSpace(ZZ, 2)
f = DynamicalSystem_affine([x^2 - 2, y^2])
type(f.as_scheme_morphism())
>>> from sage.all import *
>>> A = AffineSpace(ZZ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> f = DynamicalSystem_affine([x**Integer(2) - Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space'>

A.<x,y> = AffineSpace(ZZ, 2)
f = DynamicalSystem_affine([x^2 - 2, y^2])
type(f.as_scheme_morphism())
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: f = DynamicalSystem_affine([x^2 - 2, y^2])
sage: type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_field'>

>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> f = DynamicalSystem_affine([x**Integer(2) - Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_field'>

A.<x,y> = AffineSpace(QQ, 2)
f = DynamicalSystem_affine([x^2 - 2, y^2])
type(f.as_scheme_morphism())
>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> f = DynamicalSystem_affine([x**Integer(2) - Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_field'>

A.<x,y> = AffineSpace(QQ, 2)
f = DynamicalSystem_affine([x^2 - 2, y^2])
type(f.as_scheme_morphism())
sage: A.<x,y> = AffineSpace(GF(3), 2)
sage: f = DynamicalSystem_affine([x^2 - 2, y^2])
sage: type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_finite_field'>

>>> from sage.all import *
>>> A = AffineSpace(GF(Integer(3)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> f = DynamicalSystem_affine([x**Integer(2) - Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_finite_field'>

A.<x,y> = AffineSpace(GF(3), 2)
f = DynamicalSystem_affine([x^2 - 2, y^2])
type(f.as_scheme_morphism())
>>> from sage.all import *
>>> A = AffineSpace(GF(Integer(3)), Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
>>> f = DynamicalSystem_affine([x**Integer(2) - Integer(2), y**Integer(2)])
>>> type(f.as_scheme_morphism())
<class 'sage.schemes.affine.affine_morphism.SchemeMorphism_polynomial_affine_space_finite_field'>

A.<x,y> = AffineSpace(GF(3), 2)
f = DynamicalSystem_affine([x^2 - 2, y^2])
type(f.as_scheme_morphism())
change_ring(R, check=True)[source]

Return a new dynamical system which is this map coerced to R.

If check is True, then the initialization checks are performed.

INPUT:

  • R – ring or morphism

OUTPUT:

A new DynamicalSystem_projective that is this map coerced to R.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
sage: f = DynamicalSystem_projective([3*x^2, y^2])
sage: f.change_ring(GF(5))
Dynamical System of Projective Space of dimension 1 over Finite Field of size 5
  Defn: Defined on coordinates by sending (x : y) to (-2*x^2 : y^2)

>>> from sage.all import *
>>> P = ProjectiveSpace(ZZ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([Integer(3)*x**Integer(2), y**Integer(2)])
>>> f.change_ring(GF(Integer(5)))
Dynamical System of Projective Space of dimension 1 over Finite Field of size 5
  Defn: Defined on coordinates by sending (x : y) to (-2*x^2 : y^2)

P.<x,y> = ProjectiveSpace(ZZ, 1)
f = DynamicalSystem_projective([3*x^2, y^2])
f.change_ring(GF(5))
field_of_definition_critical(return_embedding=False, simplify_all=False, names='a')[source]

Return smallest extension of the base field which contains the critical points.

Ambient space of dynamical system must be either the affine line or projective line over a number field or finite field.

INPUT:

  • return_embedding – boolean (default: False); if True, return an embedding of base field of dynamical system into the returned number field or finite field. Note that computing this embedding might be expensive.

  • simplify_all – boolean (default: False); if True, simplify intermediate fields and also the resulting number field. Note that this is not implemented for finite fields and has no effect.

  • names – (optional) string to be used as generator for returned number field or finite field

OUTPUT:

If return_embedding is False, the field of definition as an absolute number field or finite field. If return_embedding is True, a tuple (K, phi) where phi is an embedding of the base field in K.

EXAMPLES:

Note that the number of critical points is \(2d-2\), but \((1:0)\) has multiplicity 2 in this case:

sage: # needs sage.libs.singular sage.rings.number_field
sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem([1/3*x^3 + x*y^2, y^3], domain=P)
sage: f.critical_points()
[(1 : 0)]
sage: N.<a> = f.field_of_definition_critical(); N
Number Field in a with defining polynomial x^2 + 1
sage: g = f.change_ring(N)
sage: g.critical_points()
[(-a : 1), (a : 1), (1 : 0)]

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> P = ProjectiveSpace(QQ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem([Integer(1)/Integer(3)*x**Integer(3) + x*y**Integer(2), y**Integer(3)], domain=P)
>>> f.critical_points()
[(1 : 0)]
>>> N = f.field_of_definition_critical(names=('a',)); (a,) = N._first_ngens(1); N
Number Field in a with defining polynomial x^2 + 1
>>> g = f.change_ring(N)
>>> g.critical_points()
[(-a : 1), (a : 1), (1 : 0)]

# needs sage.libs.singular sage.rings.number_field
P.<x,y> = ProjectiveSpace(QQ, 1)
f = DynamicalSystem([1/3*x^3 + x*y^2, y^3], domain=P)
f.critical_points()
N.<a> = f.field_of_definition_critical(); N
g = f.change_ring(N)
g.critical_points()
sage: # needs sage.libs.singular sage.rings.number_field
sage: A.<z> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem([z^4 + 2*z^2 + 2], domain=A)
sage: K.<a> = f.field_of_definition_critical(); K
Number Field in a with defining polynomial z^2 + 1

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> A = AffineSpace(QQ, Integer(1), names=('z',)); (z,) = A._first_ngens(1)
>>> f = DynamicalSystem([z**Integer(4) + Integer(2)*z**Integer(2) + Integer(2)], domain=A)
>>> K = f.field_of_definition_critical(names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial z^2 + 1

# needs sage.libs.singular sage.rings.number_field
A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem([z^4 + 2*z^2 + 2], domain=A)
K.<a> = f.field_of_definition_critical(); K
>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> A = AffineSpace(QQ, Integer(1), names=('z',)); (z,) = A._first_ngens(1)
>>> f = DynamicalSystem([z**Integer(4) + Integer(2)*z**Integer(2) + Integer(2)], domain=A)
>>> K = f.field_of_definition_critical(names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial z^2 + 1

# needs sage.libs.singular sage.rings.number_field
A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem([z^4 + 2*z^2 + 2], domain=A)
K.<a> = f.field_of_definition_critical(); K
sage: # needs sage.libs.singular sage.rings.finite_rings
sage: G.<a> = GF(9)
sage: R.<z> = G[]
sage: R.irreducible_element(3, algorithm='first_lexicographic')
z^3 + (a + 1)*z + a
sage: A.<x> = AffineSpace(G,1)
sage: f = DynamicalSystem([x^4 + (2*a+2)*x^2 + a*x], domain=A)
sage: f[0].derivative(x).univariate_polynomial().is_irreducible()
True
sage: f.field_of_definition_critical(return_embedding=True, names='b')
(Finite Field in b of size 3^6,
 Ring morphism:
   From: Finite Field in a of size 3^2
   To:   Finite Field in b of size 3^6
   Defn: a |--> 2*b^5 + 2*b^3 + b^2 + 2*b + 2)

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.finite_rings
>>> G = GF(Integer(9), names=('a',)); (a,) = G._first_ngens(1)
>>> R = G['z']; (z,) = R._first_ngens(1)
>>> R.irreducible_element(Integer(3), algorithm='first_lexicographic')
z^3 + (a + 1)*z + a
>>> A = AffineSpace(G,Integer(1), names=('x',)); (x,) = A._first_ngens(1)
>>> f = DynamicalSystem([x**Integer(4) + (Integer(2)*a+Integer(2))*x**Integer(2) + a*x], domain=A)
>>> f[Integer(0)].derivative(x).univariate_polynomial().is_irreducible()
True
>>> f.field_of_definition_critical(return_embedding=True, names='b')
(Finite Field in b of size 3^6,
 Ring morphism:
   From: Finite Field in a of size 3^2
   To:   Finite Field in b of size 3^6
   Defn: a |--> 2*b^5 + 2*b^3 + b^2 + 2*b + 2)

# needs sage.libs.singular sage.rings.finite_rings
G.<a> = GF(9)
R.<z> = G[]
R.irreducible_element(3, algorithm='first_lexicographic')
A.<x> = AffineSpace(G,1)
f = DynamicalSystem([x^4 + (2*a+2)*x^2 + a*x], domain=A)
f[0].derivative(x).univariate_polynomial().is_irreducible()
f.field_of_definition_critical(return_embedding=True, names='b')
>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.finite_rings
>>> G = GF(Integer(9), names=('a',)); (a,) = G._first_ngens(1)
>>> R = G['z']; (z,) = R._first_ngens(1)
>>> R.irreducible_element(Integer(3), algorithm='first_lexicographic')
z^3 + (a + 1)*z + a
>>> A = AffineSpace(G,Integer(1), names=('x',)); (x,) = A._first_ngens(1)
>>> f = DynamicalSystem([x**Integer(4) + (Integer(2)*a+Integer(2))*x**Integer(2) + a*x], domain=A)
>>> f[Integer(0)].derivative(x).univariate_polynomial().is_irreducible()
True
>>> f.field_of_definition_critical(return_embedding=True, names='b')
(Finite Field in b of size 3^6,
 Ring morphism:
   From: Finite Field in a of size 3^2
   To:   Finite Field in b of size 3^6
   Defn: a |--> 2*b^5 + 2*b^3 + b^2 + 2*b + 2)

# needs sage.libs.singular sage.rings.finite_rings
G.<a> = GF(9)
R.<z> = G[]
R.irreducible_element(3, algorithm='first_lexicographic')
A.<x> = AffineSpace(G,1)
f = DynamicalSystem([x^4 + (2*a+2)*x^2 + a*x], domain=A)
f[0].derivative(x).univariate_polynomial().is_irreducible()
f.field_of_definition_critical(return_embedding=True, names='b')
field_of_definition_periodic(n, formal=False, return_embedding=False, simplify_all=False, names='a')[source]

Return smallest extension of the base field which contains all fixed points of the n-th iterate

Ambient space of dynamical system must be either the affine line or projective line over a number field or finite field.

INPUT:

  • n – positive integer

  • formal – boolean (default: False); True signals to return number field or finite field over which the formal periodic points are defined, where a formal periodic point is a root of the n-th dynatomic polynomial. False specifies to find number field or finite field over which all periodic points of the n-th iterate are defined.

  • return_embedding – boolean (default: False); if True, return an embedding of base field of dynamical system into the returned number field or finite field. Note that computing this embedding might be expensive.

  • simplify_all – boolean (default: False); if True, simplify intermediate fields and also the resulting number field. Note that this is not implemented for finite fields and has no effect.

  • names – (optional) string to be used as generator for returned number field or finite field

OUTPUT:

If return_embedding is False, the field of definition as an absolute number field or finite field. If return_embedding is True, a tuple (K, phi) where phi is an embedding of the base field in K.

EXAMPLES:

sage: # needs sage.libs.singular sage.rings.number_field
sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem([x^2, y^2], domain=P)
sage: f.periodic_points(3, minimal=False)
[(0 : 1), (1 : 0), (1 : 1)]
sage: N.<a> = f.field_of_definition_periodic(3); N
Number Field in a with defining polynomial x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
sage: sorted(f.periodic_points(3, minimal=False, R=N), key=str)
[(-a^5 - a^4 - a^3 - a^2 - a - 1 : 1),
 (0 : 1),
 (1 : 0),
 (1 : 1),
 (a : 1),
 (a^2 : 1),
 (a^3 : 1),
 (a^4 : 1),
 (a^5 : 1)]

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> P = ProjectiveSpace(QQ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem([x**Integer(2), y**Integer(2)], domain=P)
>>> f.periodic_points(Integer(3), minimal=False)
[(0 : 1), (1 : 0), (1 : 1)]
>>> N = f.field_of_definition_periodic(Integer(3), names=('a',)); (a,) = N._first_ngens(1); N
Number Field in a with defining polynomial x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
>>> sorted(f.periodic_points(Integer(3), minimal=False, R=N), key=str)
[(-a^5 - a^4 - a^3 - a^2 - a - 1 : 1),
 (0 : 1),
 (1 : 0),
 (1 : 1),
 (a : 1),
 (a^2 : 1),
 (a^3 : 1),
 (a^4 : 1),
 (a^5 : 1)]

# needs sage.libs.singular sage.rings.number_field
P.<x,y> = ProjectiveSpace(QQ, 1)
f = DynamicalSystem([x^2, y^2], domain=P)
f.periodic_points(3, minimal=False)
N.<a> = f.field_of_definition_periodic(3); N
sorted(f.periodic_points(3, minimal=False, R=N), key=str)
sage: # needs sage.libs.singular sage.rings.number_field
sage: A.<z> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem([(z^2 + 1)/(2*z + 1)], domain=A)
sage: K.<a> = f.field_of_definition_periodic(2); K
Number Field in a with defining polynomial z^4 + 12*z^3 + 39*z^2 + 18*z + 171
sage: F.<b> = f.field_of_definition_periodic(2, formal=True); F
Number Field in b with defining polynomial z^2 + 3*z + 6

>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> A = AffineSpace(QQ, Integer(1), names=('z',)); (z,) = A._first_ngens(1)
>>> f = DynamicalSystem([(z**Integer(2) + Integer(1))/(Integer(2)*z + Integer(1))], domain=A)
>>> K = f.field_of_definition_periodic(Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial z^4 + 12*z^3 + 39*z^2 + 18*z + 171
>>> F = f.field_of_definition_periodic(Integer(2), formal=True, names=('b',)); (b,) = F._first_ngens(1); F
Number Field in b with defining polynomial z^2 + 3*z + 6

# needs sage.libs.singular sage.rings.number_field
A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem([(z^2 + 1)/(2*z + 1)], domain=A)
K.<a> = f.field_of_definition_periodic(2); K
F.<b> = f.field_of_definition_periodic(2, formal=True); F
>>> from sage.all import *
>>> # needs sage.libs.singular sage.rings.number_field
>>> A = AffineSpace(QQ, Integer(1), names=('z',)); (z,) = A._first_ngens(1)
>>> f = DynamicalSystem([(z**Integer(2) + Integer(1))/(Integer(2)*z + Integer(1))], domain=A)
>>> K = f.field_of_definition_periodic(Integer(2), names=('a',)); (a,) = K._first_ngens(1); K
Number Field in a with defining polynomial z^4 + 12*z^3 + 39*z^2 + 18*z + 171
>>> F = f.field_of_definition_periodic(Integer(2), formal=True, names=('b',)); (b,) = F._first_ngens(1); F
Number Field in b with defining polynomial z^2 + 3*z + 6

# needs sage.libs.singular sage.rings.number_field
A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem([(z^2 + 1)/(2*z + 1)], domain=A)
K.<a> = f.field_of_definition_periodic(2); K
F.<b> = f.field_of_definition_periodic(2, formal=True); F
sage: # needs sage.rings.finite_rings
sage: G.<a> = GF(4)
sage: A.<x> = AffineSpace(G, 1)
sage: f = DynamicalSystem([x^2 + (a+1)*x + 1], domain=A)
sage: g = f.nth_iterate_map(2)[0]
sage: (g-x).univariate_polynomial().factor()
(x + 1) * (x + a + 1) * (x^2 + a*x + 1)
sage: f.field_of_definition_periodic(2, return_embedding=True, names='b')
(Finite Field in b of size 2^4,
 Ring morphism:
   From: Finite Field in a of size 2^2
   To:   Finite Field in b of size 2^4
   Defn: a |--> b^2 + b)

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> G = GF(Integer(4), names=('a',)); (a,) = G._first_ngens(1)
>>> A = AffineSpace(G, Integer(1), names=('x',)); (x,) = A._first_ngens(1)
>>> f = DynamicalSystem([x**Integer(2) + (a+Integer(1))*x + Integer(1)], domain=A)
>>> g = f.nth_iterate_map(Integer(2))[Integer(0)]
>>> (g-x).univariate_polynomial().factor()
(x + 1) * (x + a + 1) * (x^2 + a*x + 1)
>>> f.field_of_definition_periodic(Integer(2), return_embedding=True, names='b')
(Finite Field in b of size 2^4,
 Ring morphism:
   From: Finite Field in a of size 2^2
   To:   Finite Field in b of size 2^4
   Defn: a |--> b^2 + b)

# needs sage.rings.finite_rings
G.<a> = GF(4)
A.<x> = AffineSpace(G, 1)
f = DynamicalSystem([x^2 + (a+1)*x + 1], domain=A)
g = f.nth_iterate_map(2)[0]
(g-x).univariate_polynomial().factor()
f.field_of_definition_periodic(2, return_embedding=True, names='b')
>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> G = GF(Integer(4), names=('a',)); (a,) = G._first_ngens(1)
>>> A = AffineSpace(G, Integer(1), names=('x',)); (x,) = A._first_ngens(1)
>>> f = DynamicalSystem([x**Integer(2) + (a+Integer(1))*x + Integer(1)], domain=A)
>>> g = f.nth_iterate_map(Integer(2))[Integer(0)]
>>> (g-x).univariate_polynomial().factor()
(x + 1) * (x + a + 1) * (x^2 + a*x + 1)
>>> f.field_of_definition_periodic(Integer(2), return_embedding=True, names='b')
(Finite Field in b of size 2^4,
 Ring morphism:
   From: Finite Field in a of size 2^2
   To:   Finite Field in b of size 2^4
   Defn: a |--> b^2 + b)

# needs sage.rings.finite_rings
G.<a> = GF(4)
A.<x> = AffineSpace(G, 1)
f = DynamicalSystem([x^2 + (a+1)*x + 1], domain=A)
g = f.nth_iterate_map(2)[0]
(g-x).univariate_polynomial().factor()
f.field_of_definition_periodic(2, return_embedding=True, names='b')
field_of_definition_preimage(point, n, return_embedding=False, simplify_all=False, names='a')[source]

Return smallest extension of the base field which contains the n-th preimages of point

Ambient space of dynamical system must be either the affine line or projective line over a number field or finite field.

INPUT:

  • point – a point in this map’s domain

  • n – positive integer

  • return_embedding – boolean (default: False); if True, return an embedding of base field of dynamical system into the returned number field or finite field. Note that computing this embedding might be expensive.

  • simplify_all – boolean (default: False); if True, simplify intermediate fields and also the resulting number field. Note that this is not implemented for finite fields and has no effect.

  • names – (optional) string to be used as generator for returned number field or finite field

OUTPUT:

If return_embedding is False, the field of definition as an absolute number field or finite field. If return_embedding is True, a tuple (K, phi) where phi is an embedding of the base field in K.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: f = DynamicalSystem([1/3*x^2 + 2/3*x*y, x^2 - 2*y^2], domain=P)
sage: N.<a> = f.field_of_definition_preimage(P(1,1), 2,                     # needs sage.rings.number_field
....:                                        simplify_all=True); N
Number Field in a with defining polynomial
 x^8 - 4*x^7 - 128*x^6 + 398*x^5 + 3913*x^4 - 8494*x^3 - 26250*x^2 + 30564*x - 2916

>>> from sage.all import *
>>> P = ProjectiveSpace(QQ, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem([Integer(1)/Integer(3)*x**Integer(2) + Integer(2)/Integer(3)*x*y, x**Integer(2) - Integer(2)*y**Integer(2)], domain=P)
>>> N = f.field_of_definition_preimage(P(Integer(1),Integer(1)), Integer(2),                     # needs sage.rings.number_field
...                                        simplify_all=True, names=('a',)); (a,) = N._first_ngens(1); N
Number Field in a with defining polynomial
 x^8 - 4*x^7 - 128*x^6 + 398*x^5 + 3913*x^4 - 8494*x^3 - 26250*x^2 + 30564*x - 2916

P.<x,y> = ProjectiveSpace(QQ, 1)
f = DynamicalSystem([1/3*x^2 + 2/3*x*y, x^2 - 2*y^2], domain=P)
N.<a> = f.field_of_definition_preimage(P(1,1), 2,                     # needs sage.rings.number_field
                                       simplify_all=True); N
sage: A.<z> = AffineSpace(QQ, 1)
sage: f = DynamicalSystem([z^2], domain=A)
sage: K.<a> = f.field_of_definition_preimage(A(1), 3); K                    # needs sage.rings.number_field
Number Field in a with defining polynomial z^4 + 1

>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(1), names=('z',)); (z,) = A._first_ngens(1)
>>> f = DynamicalSystem([z**Integer(2)], domain=A)
>>> K = f.field_of_definition_preimage(A(Integer(1)), Integer(3), names=('a',)); (a,) = K._first_ngens(1); K                    # needs sage.rings.number_field
Number Field in a with defining polynomial z^4 + 1

A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem([z^2], domain=A)
K.<a> = f.field_of_definition_preimage(A(1), 3); K                    # needs sage.rings.number_field
>>> from sage.all import *
>>> A = AffineSpace(QQ, Integer(1), names=('z',)); (z,) = A._first_ngens(1)
>>> f = DynamicalSystem([z**Integer(2)], domain=A)
>>> K = f.field_of_definition_preimage(A(Integer(1)), Integer(3), names=('a',)); (a,) = K._first_ngens(1); K                    # needs sage.rings.number_field
Number Field in a with defining polynomial z^4 + 1

A.<z> = AffineSpace(QQ, 1)
f = DynamicalSystem([z^2], domain=A)
K.<a> = f.field_of_definition_preimage(A(1), 3); K                    # needs sage.rings.number_field
sage: G = GF(5)
sage: P.<x,y> = ProjectiveSpace(G, 1)
sage: f = DynamicalSystem([x^2 + 2*y^2, y^2], domain=P)
sage: f.field_of_definition_preimage(P(2,1), 2, return_embedding=True,      # needs sage.rings.number_field
....:                                names='a')
(Finite Field in a of size 5^2,
 Ring morphism:
   From: Finite Field of size 5
   To:   Finite Field in a of size 5^2
   Defn: 1 |--> 1)

>>> from sage.all import *
>>> G = GF(Integer(5))
>>> P = ProjectiveSpace(G, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem([x**Integer(2) + Integer(2)*y**Integer(2), y**Integer(2)], domain=P)
>>> f.field_of_definition_preimage(P(Integer(2),Integer(1)), Integer(2), return_embedding=True,      # needs sage.rings.number_field
...                                names='a')
(Finite Field in a of size 5^2,
 Ring morphism:
   From: Finite Field of size 5
   To:   Finite Field in a of size 5^2
   Defn: 1 |--> 1)

G = GF(5)
P.<x,y> = ProjectiveSpace(G, 1)
f = DynamicalSystem([x^2 + 2*y^2, y^2], domain=P)
f.field_of_definition_preimage(P(2,1), 2, return_embedding=True,      # needs sage.rings.number_field
                               names='a')
>>> from sage.all import *
>>> G = GF(Integer(5))
>>> P = ProjectiveSpace(G, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem([x**Integer(2) + Integer(2)*y**Integer(2), y**Integer(2)], domain=P)
>>> f.field_of_definition_preimage(P(Integer(2),Integer(1)), Integer(2), return_embedding=True,      # needs sage.rings.number_field
...                                names='a')
(Finite Field in a of size 5^2,
 Ring morphism:
   From: Finite Field of size 5
   To:   Finite Field in a of size 5^2
   Defn: 1 |--> 1)

G = GF(5)
P.<x,y> = ProjectiveSpace(G, 1)
f = DynamicalSystem([x^2 + 2*y^2, y^2], domain=P)
f.field_of_definition_preimage(P(2,1), 2, return_embedding=True,      # needs sage.rings.number_field
                               names='a')
specialization(D=None, phi=None, homset=None)[source]

Specialization of this dynamical system.

Given a family of maps defined over a polynomial ring. A specialization is a particular member of that family. The specialization can be specified either by a dictionary or a SpecializationMorphism.

INPUT:

  • D – (optional) dictionary

  • phi – (optional) SpecializationMorphism

  • homset – (optional) homset of specialized map

OUTPUT: DynamicalSystem

EXAMPLES:

sage: R.<c> = PolynomialRing(QQ)
sage: P.<x,y> = ProjectiveSpace(R, 1)
sage: f = DynamicalSystem_projective([x^2 + c*y^2, y^2], domain=P)
sage: f.specialization({c:1})
Dynamical System of Projective Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x : y) to (x^2 + y^2 : y^2)

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('c',)); (c,) = R._first_ngens(1)
>>> P = ProjectiveSpace(R, Integer(1), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> f = DynamicalSystem_projective([x**Integer(2) + c*y**Integer(2), y**Integer(2)], domain=P)
>>> f.specialization({c:Integer(1)})
Dynamical System of Projective Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x : y) to (x^2 + y^2 : y^2)

R.<c> = PolynomialRing(QQ)
P.<x,y> = ProjectiveSpace(R, 1)
f = DynamicalSystem_projective([x^2 + c*y^2, y^2], domain=P)
f.specialization({c:1})