# Morphisms on projective varieties¶

A morphism of schemes determined by rational functions that define what the morphism does on points in the ambient projective space.

AUTHORS:

• David Kohel, William Stein
• William Stein (2006-02-11): fixed bug where P(0,0,0) was allowed as a projective point.
• Volker Braun (2011-08-08): Renamed classes, more documentation, misc cleanups.
• Ben Hutz (2013-03) iteration functionality and new directory structure for affine/projective, height functionality
• Brian Stout, Ben Hutz (Nov 2013) - added minimal model functionality
• Dillon Rose (2014-01): Speed enhancements
• Ben Hutz (2015-11): iteration of subschemes
class sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space(parent, polys, check=True)

A morphism of schemes determined by rational functions that define what the morphism does on points in the ambient projective space.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: P1 = ProjectiveSpace(R)
sage: H = P1.Hom(P1)
sage: H([y,2*x])
Scheme endomorphism of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(y : 2*x)


An example of a morphism between projective plane curves (see trac ticket #10297):

sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: f = x^3+y^3+60*z^3
sage: g = y^2*z-( x^3 - 6400*z^3/3)
sage: C = Curve(f)
sage: E = Curve(g)
sage: xbar,ybar,zbar = C.coordinate_ring().gens()
sage: H = C.Hom(E)
sage: H([zbar,xbar-ybar,-(xbar+ybar)/80])
Scheme morphism:
From: Projective Plane Curve over Rational Field defined by x^3 + y^3 + 60*z^3
To:   Projective Plane Curve over Rational Field defined by -x^3 + y^2*z + 6400/3*z^3
Defn: Defined on coordinates by sending (x : y : z) to
(z : x - y : -1/80*x - 1/80*y)


A more complicated example:

sage: P2.<x,y,z> = ProjectiveSpace(2, QQ)
sage: P1 = P2.subscheme(x-y)
sage: H12 = P1.Hom(P2)
sage: H12([x^2, x*z, z^2])
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x - y
To:   Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(x^2 : x*z : z^2)


We illustrate some error checking:

sage: R.<x,y> = QQ[]
sage: P1 = ProjectiveSpace(R)
sage: H = P1.Hom(P1)
sage: f = H([x-y, x*y])
Traceback (most recent call last):
...
ValueError: polys (=[x - y, x*y]) must be of the same degree

sage: H([x-1, x*y+x])
Traceback (most recent call last):
...
ValueError: polys (=[x - 1, x*y + x]) must be homogeneous

sage: H([exp(x),exp(y)])
Traceback (most recent call last):
...
TypeError: polys (=[e^x, e^y]) must be elements of
Multivariate Polynomial Ring in x, y over Rational Field


We can also compute the forward image of subschemes through elimination. In particular, let $$X = V(h_1,\ldots, h_t)$$ and define the ideal $$I = (h_1,\ldots,h_t,y_0-f_0(\bar{x}), \ldots, y_n-f_n(\bar{x}))$$. Then the elimination ideal $$I_{n+1} = I \cap K[y_0,\ldots,y_n]$$ is a homogeneous ideal and $$f(X) = V(I_{n+1})$$:

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: H = End(P)
sage: f = H([(x-2*y)^2, (x-2*z)^2, x^2])
sage: X = P.subscheme(y-z)
sage: f(f(f(X)))
Closed subscheme of Projective Space of dimension 2 over Rational Field
defined by:
y - z

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(P.subscheme([x,y,z]))
Closed subscheme of Projective Space of dimension 3 over Rational Field
defined by:
w,
y,
x

as_dynamical_system()

Return this endomorphism as a DynamicalSystem_projective.

OUTPUT:

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: H = End(P)
sage: f = H([x^2, y^2, z^2])
sage: type(f.as_dynamical_system())
<class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective'>

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([x^2-y^2, y^2])
sage: type(f.as_dynamical_system())
<class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective_field'>

sage: P.<x,y> = ProjectiveSpace(GF(5), 1)
sage: H = End(P)
sage: f = H([x^2, y^2])
sage: type(f.as_dynamical_system())
<class 'sage.dynamics.arithmetic_dynamics.projective_ds.DynamicalSystem_projective_finite_field'>

sage: P.<x,y> = ProjectiveSpace(RR, 1)
sage: f = DynamicalSystem([x^2 + y^2, y^2], P)
sage: g = f.as_dynamical_system()
sage: g is f
True

automorphism_group(**kwds)

Return the automorphism group.

EXAMPLES:

sage: R.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(R)
sage: f = H([x^2-y^2, x*y])
sage: f.automorphism_group(return_functions=True)
doctest:warning
...
[x, -x]

canonical_height(P, **kwds)

Return the canonical height of the point.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(ZZ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, 2*x*y]);
sage: f.canonical_height(P.point([5,4]), error_bound=0.001)
doctest:warning
...
2.1970553519503404898926835324

conjugate(M)

Return the conjugate of this map.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(ZZ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, y^2])
sage: f.conjugate(matrix([[1,2], [0,1]]))
doctest:warning
...
Dynamical System of Projective Space of dimension 1 over Integer Ring
Defn: Defined on coordinates by sending (x : y) to
(x^2 + 4*x*y + 3*y^2 : y^2)

critical_height(**kwds)

Return the critical height.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f =H([x^3+7*y^3, 11*y^3])
sage: f.critical_height()
doctest:warning
...
1.1989273321156851418802151128

critical_point_portrait(check=True, embedding=None)

Return the directed graph of critical point portrait.

EXAMPLES:

sage: R.<z> = QQ[]
sage: K.<v> = NumberField(z^6 + 2*z^5 + 2*z^4 + 2*z^3 + z^2 + 1)
sage: PS.<x,y> = ProjectiveSpace(K,1)
sage: H = End(PS)
sage: f = H([x^2+v*y^2, y^2])
sage: f.critical_point_portrait(check=False, embedding=K.embeddings(QQbar)[0]) # long time
doctest:warning
...
Looped digraph on 6 vertices

critical_points(R=None)

Return the critical points.

Examples:

sage: set_verbose(None)
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^3-2*x*y^2 + 2*y^3, y^3])
sage: f.critical_points()
doctest:warning
...
[(1 : 0)]

critical_subscheme()

Return the critical subscheme.

Examples:

sage: set_verbose(None)
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^3-2*x*y^2 + 2*y^3, y^3])
sage: f.critical_subscheme()
doctest:warning
...
Closed subscheme of Projective Space of dimension 1 over Rational Field
defined by:
9*x^2*y^2 - 6*y^4

degree()

Return the degree of this map.

The degree is defined as the degree of the homogeneous polynomials that are the coordinates of this map.

OUTPUT:

• A positive integer

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([x^2+y^2, y^2])
sage: f.degree()
2

sage: P.<x,y,z> = ProjectiveSpace(CC,2)
sage: H = Hom(P,P)
sage: f = H([x^3+y^3, y^2*z, z*x*y])
sage: f.degree()
3

sage: R.<t> = PolynomialRing(QQ)
sage: P.<x,y,z> = ProjectiveSpace(R,2)
sage: H = Hom(P,P)
sage: f = H([x^2+t*y^2, (2-t)*y^2, z^2])
sage: f.degree()
2

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.degree()
2

degree_sequence(iterates=2)

Return the sequence of degrees of iterates.

EXAMPLES:

sage: P2.<X,Y,Z> = ProjectiveSpace(QQ, 2)
sage: H = End(P2)
sage: f = H([Z^2, X*Y, Y^2])
sage: f.degree_sequence(15)
doctest:warning
...
[2, 3, 5, 8, 11, 17, 24, 31, 45, 56, 68, 91, 93, 184, 275]

dehomogenize(n)

Returns the standard dehomogenization at the n[0] coordinate for the domain and the n[1] coordinate for the codomain.

Note that the new function is defined over the fraction field of the base ring of this map.

INPUT:

• n – a tuple of nonnegative integers. If n is an integer, then the two values of
the tuple are assumed to be the same.

OUTPUT:

• SchemeMorphism_polynomial_affine_space.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(ZZ,1)
sage: H = Hom(P,P)
sage: f = H([x^2+y^2, y^2])
sage: f.dehomogenize(0)
Scheme endomorphism of Affine Space of dimension 1 over Integer Ring
Defn: Defined on coordinates by sending (y) to
(y^2/(y^2 + 1))

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([x^2-y^2, y^2])
sage: f.dehomogenize((0,1))
Scheme morphism:
From: Affine Space of dimension 1 over Rational Field
To:   Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (y) to
((-y^2 + 1)/y^2)

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = Hom(P,P)
sage: f = H([x^2+y^2, y^2-z^2, 2*z^2])
sage: f.dehomogenize(2)
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x, y) to
(1/2*x^2 + 1/2*y^2, 1/2*y^2 - 1/2)

sage: R.<t> = PolynomialRing(QQ)
sage: P.<x,y,z> = ProjectiveSpace(FractionField(R),2)
sage: H = Hom(P,P)
sage: f = H([x^2+t*y^2, t*y^2-z^2, t*z^2])
sage: f.dehomogenize(2)
Scheme endomorphism of Affine Space of dimension 2 over Fraction Field
of Univariate Polynomial Ring in t over Rational Field
Defn: Defined on coordinates by sending (x, y) to
(1/t*x^2 + y^2, y^2 - 1/t)

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, x*z])
sage: f.dehomogenize(2)
Scheme endomorphism of Closed subscheme of Affine Space of dimension 2 over Integer Ring defined by:
x^2 - y^2
Defn: Defined on coordinates by sending (x, y) to
(x, y^2/x)

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2 - 2*x*y, y^2])
sage: f.dehomogenize(0).homogenize(0) == f
True

sage: K.<w> = QuadraticField(3)
sage: O = K.ring_of_integers()
sage: P.<x,y> = ProjectiveSpace(O,1)
sage: H = End(P)
sage: f = H([x^2 - O(w)*y^2,y^2])
sage: f.dehomogenize(1)
Scheme endomorphism of Affine Space of dimension 1 over Maximal Order in Number Field in w with defining polynomial x^2 - 3
Defn: Defined on coordinates by sending (x) to
(x^2 - w)

dynamical_degree(N=3, prec=53)

Return the dynamical degree.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([x^2 + (x*y), y^2])
sage: f.dynamical_degree()
doctest:warning
...
2.00000000000000

dynatomic_polynomial(period)

Return the dynatomic polynomial.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2 + y^2, y^2])
sage: f.dynatomic_polynomial(2)
doctest:warning
...
x^2 + x*y + 2*y^2

global_height(prec=None)

Returns the maximum of the absolute logarithmic heights of the coefficients in any of the coordinate functions of this map.

INPUT:

• prec – desired floating point precision (default: default RealField precision).

OUTPUT:

• a real number.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([1/1331*x^2+1/4000*y^2, 210*x*y]);
sage: f.global_height()
8.29404964010203


This function does not automatically normalize:

sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
sage: H = Hom(P,P)
sage: f = H([4*x^2+100*y^2, 210*x*y, 10000*z^2]);
sage: f.global_height()
9.21034037197618
sage: f.normalize_coordinates()
sage: f.global_height()
8.51719319141624

sage: R.<z> = PolynomialRing(QQ)
sage: K.<w> = NumberField(z^2-2)
sage: O = K.maximal_order()
sage: P.<x,y> = ProjectiveSpace(O,1)
sage: H = Hom(P,P)
sage: f = H([2*x^2 + 3*O(w)*y^2, O(w)*y^2])
sage: f.global_height()
1.44518587894808

sage: P.<x,y> = ProjectiveSpace(QQbar,1)
sage: P2.<u,v,w> = ProjectiveSpace(QQbar,2)
sage: H = Hom(P,P2)
sage: f = H([x^2 + QQbar(I)*x*y + 3*y^2, y^2, QQbar(sqrt(5))*x*y])
sage: f.global_height()
1.09861228866811

green_function(P, v, **kwds)

Return the value of the Green’s function at the point.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, x*y]);
sage: Q = P(5, 1)
sage: f.green_function(Q, 0, N=30)
doctest:warning
...
1.6460930159932946233759277576

height_difference_bound(prec=None)

Return the bound on the difference of the height and canonical height.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, x*y])
sage: f.height_difference_bound()
doctest:warning
...
1.38629436111989

is_PGL_minimal(prime_list=None)

Return whether the representation is PGL minimal.

EXAMPLES:

sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
sage: H = End(PS)
sage: f = H([X^2+3*Y^2, X*Y])
sage: f.is_PGL_minimal()
doctest:warning
...
True

is_morphism()

returns True if this map is a morphism.

The map is a morphism if and only if the ideal generated by the defining polynomials is the unit ideal (no common zeros of the defining polynomials).

OUTPUT:

• Boolean

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([x^2+y^2, y^2])
sage: f.is_morphism()
True

sage: P.<x,y,z> = ProjectiveSpace(RR,2)
sage: H = Hom(P,P)
sage: f = H([x*z-y*z, x^2-y^2, z^2])
sage: f.is_morphism()
False

sage: R.<t> = PolynomialRing(GF(5))
sage: P.<x,y,z> = ProjectiveSpace(R,2)
sage: H = Hom(P,P)
sage: f = H([x*z-t*y^2, x^2-y^2, t*z^2])
sage: f.is_morphism()
True


Map that is not morphism on projective space, but is over a subscheme:

sage: P.<x,y,z> = ProjectiveSpace(RR,2)
sage: X = P.subscheme([x*y + y*z])
sage: H = Hom(X,X)
sage: f = H([x*z-y*z, x^2-y^2, z^2])
sage: f.is_morphism()
True

is_postcritically_finite(err=0.01, embedding=None)

Return whether the map is postcritically finite.

Examples:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2 - y^2, y^2])
sage: f.is_postcritically_finite()
doctest:warning
...
True

local_height(v, prec=None)

Returns the maximum of the local height of the coefficients in any of the coordinate functions of this map.

INPUT:

• v – a prime or prime ideal of the base ring.
• prec – desired floating point precision (default: default RealField precision).

OUTPUT:

• a real number.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([1/1331*x^2+1/4000*y^2, 210*x*y]);
sage: f.local_height(1331)
7.19368581839511


This function does not automatically normalize:

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = Hom(P,P)
sage: f = H([4*x^2+3/100*y^2, 8/210*x*y, 1/10000*z^2]);
sage: f.local_height(2)
2.77258872223978
sage: f.normalize_coordinates()
sage: f.local_height(2)
0.000000000000000

sage: R.<z> = PolynomialRing(QQ)
sage: K.<w> = NumberField(z^2-2)
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = Hom(P,P)
sage: f = H([2*x^2 + w/3*y^2, 1/w*y^2])
sage: f.local_height(K.ideal(3))
1.09861228866811

local_height_arch(i, prec=None)

Returns the maximum of the local height at the i-th infinite place of the coefficients in any of the coordinate functions of this map.

INPUT:

• i – an integer.
• prec – desired floating point precision (default: default RealField precision).

OUTPUT:

• a real number.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([1/1331*x^2+1/4000*y^2, 210*x*y]);
sage: f.local_height_arch(0)
5.34710753071747

sage: R.<z> = PolynomialRing(QQ)
sage: K.<w> = NumberField(z^2-2)
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = Hom(P,P)
sage: f = H([2*x^2 + w/3*y^2, 1/w*y^2])
sage: f.local_height_arch(1)
0.6931471805599453094172321214582

minimal_model(return_transformation=False, prime_list=None)

Return the minimal model.

EXAMPLES:

sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
sage: H = End(PS)
sage: f = H([X^2+3*Y^2, X*Y])
sage: f.minimal_model(return_transformation=True)
doctest:warning
...
(
Dynamical System of Projective Space of dimension 1 over Rational
Field
Defn: Defined on coordinates by sending (X : Y) to
(X^2 + 3*Y^2 : X*Y)
,
[1 0]
[0 1]
)

multiplier(P, n, check=True)

Return the multiplier at the point.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = End(P)
sage: f = H([x^2,y^2, 4*z^2]);
sage: Q = P.point([4,4,1], False);
sage: f.multiplier(Q,1)
doctest:warning
...
[2 0]
[0 2]

multiplier_spectra(n, formal=False, embedding=None, type='point')

Computes the formal n multiplier spectra of this map.

This is the set of multipliers of the periodic points of formal period n included with the appropriate multiplicity. User can also specify to compute the n multiplier spectra instead which includes the multipliers of all periodic points of period n.The map must be defined over projective space of dimension 1 over a number field.

The parameter type determines if the multipliers are computed one per cycle (with multiplicity) or one per point (with multiplicity). Note that in the cycle case, a map with a cycle which collapses into multiple smaller cycles will have more multipliers than one that does not.

INPUT:

• n - a positive integer, the period.
• formal - a Boolean. True specifies to find the formal n multiplier spectra
of this map. False specifies to find the n multiplier spectra of this map. Default: False.
• embedding - embedding of the base field into $$\QQbar$$.
• type - string - either point or cycle depending on whether you
compute one multiplier per point or one per cycle. Default : point.

OUTPUT:

• a list of $$\QQbar$$ elements.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([4608*x^10 - 2910096*x^9*y + 325988068*x^8*y^2 + 31825198932*x^7*y^3 - 4139806626613*x^6*y^4\
- 44439736715486*x^5*y^5 + 2317935971590902*x^4*y^6 - 15344764859590852*x^3*y^7 + 2561851642765275*x^2*y^8\
+ 113578270285012470*x*y^9 - 150049940203963800*y^10, 4608*y^10])
sage: f.multiplier_spectra(1)
doctest:warning
...
[0, -7198147681176255644585/256, 848446157556848459363/19683, -3323781962860268721722583135/35184372088832,
529278480109921/256, -4290991994944936653/2097152, 1061953534167447403/19683, -3086380435599991/9,
82911372672808161930567/8192, -119820502365680843999, 3553497751559301575157261317/8192]

normalize_coordinates()

Scales by 1/gcd of the coordinate functions.

Also, scales to clear any denominators from the coefficients. This is done in place.

OUTPUT:

• None.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([5/4*x^3, 5*x*y^2])
sage: f.normalize_coordinates(); f
Scheme endomorphism of Projective Space of dimension 1 over Rational
Field
Defn: Defined on coordinates by sending (x : y) to
(x^2 : 4*y^2)

sage: P.<x,y,z> = ProjectiveSpace(GF(7),2)
sage: X = P.subscheme(x^2-y^2)
sage: H = Hom(X,X)
sage: f = H([x^3+x*y^2, x*y^2, x*z^2])
sage: f.normalize_coordinates(); f
Scheme endomorphism of Closed subscheme of Projective Space of dimension
2 over Finite Field of size 7 defined by:
x^2 - y^2
Defn: Defined on coordinates by sending (x : y : z) to
(2*y^2 : y^2 : z^2)

sage: R.<a,b> = QQ[]
sage: P.<x,y,z> = ProjectiveSpace(R, 2)
sage: H = End(P)
sage: f = H([a*(x*z+y^2)*x^2, a*b*(x*z+y^2)*y^2, a*(x*z+y^2)*z^2])
sage: f.normalize_coordinates(); f
Scheme endomorphism of Projective Space of dimension 2 over Multivariate
Polynomial Ring in a, b over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(x^2 : b*y^2 : z^2)


Note

gcd raises an error if the base_ring does not support gcds.

nth_iterate(P, n, **kwds)

Return the nth iterate of the point.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(ZZ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, 2*y^2])
sage: Q = P(1,1)
sage: f.nth_iterate(Q,4)
doctest:warning
...
(32768 : 32768)

nth_iterate_map(n, normalize=False)

Return the nth iterate of the map.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, y^2])
sage: f.nth_iterate_map(2)
doctest:warning
...
Dynamical System of Projective Space of dimension 1 over Rational
Field
Defn: Defined on coordinates by sending (x : y) to
(x^4 + 2*x^2*y^2 + 2*y^4 : y^4)

orbit(P, N, **kwds)

Return the orbit of the point.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
sage: H = End(P)
sage: f = H([x^2+y^2, y^2-z^2, 2*z^2])
sage: f.orbit(P(1,2,1), 3)
doctest:warning
...
[(1 : 2 : 1), (5 : 3 : 2), (34 : 5 : 8), (1181 : -39 : 128)]

periodic_points(n, minimal=True, R=None, algorithm='variety', return_scheme=False)

Return the periodic points.

EXAMPLES:

sage: set_verbose(None)
sage: P.<x,y> = ProjectiveSpace(QQbar,1)
sage: H = End(P)
sage: f = H([x^2-x*y+y^2, x^2-y^2+x*y])
sage: f.periodic_points(1)
doctest:warning
...
[(-0.500000000000000? - 0.866025403784439?*I : 1), (-0.500000000000000? + 0.866025403784439?*I : 1),
(1 : 1)]

possible_periods(**kwds)

Return the possible periods of a rational periodic point.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2-29/16*y^2, y^2])
sage: f.possible_periods(ncpus=1)
doctest:warning
...
[1, 3]

primes_of_bad_reduction(check=True)

Return the primes of bad reduction.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([1/3*x^2+1/2*y^2, y^2])
doctest:warning
...
[2, 3]

reduced_form(prec=300, return_conjugation=True, error_limit=1e-06)

Return the reduced form of the map.

EXAMPLES:

sage: PS.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(PS)
sage: f = H([x^3, 10794303*x^3 + 146526*x^2*y + 663*x*y^2 + y^3])
sage: f.reduced_form()
doctest:warning
...
(
Dynamical System of Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(x^3 + 3*x*y^2 : y^3) ,

[  0  -1]
[  1 221]
)

resultant(normalize=False)

Computes the resultant of the defining polynomials of this dynamical system.

If normalize is True, then first normalize the coordinate functions with normalize_coordinates(). Map must be an endomorphism

INPUT:

• normalize – Boolean (optional - default: False).

OUTPUT: an element of the base ring of this map.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2+y^2, 6*y^2])
sage: f.resultant()
doctest:warning
...
36

sage: P2.<x,y,z> = ProjectiveSpace(QQ,2)
sage: P1.<u,v> = ProjectiveSpace(QQ,1)
sage: H = Hom(P2,P1)
sage: f = H([x,y])
sage: f.resultant()
Traceback (most recent call last):
...
ValueError: must be an endomorphism

scale_by(t)

Scales each coordinate by a factor of t.

A TypeError occurs if the point is not in the coordinate_ring of the parent after scaling.

INPUT:

• t – a ring element.

OUTPUT:

• None.

EXAMPLES:

sage: A.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(A,A)
sage: f = H([x^3-2*x*y^2,x^2*y])
sage: f.scale_by(1/x)
sage: f
Scheme endomorphism of Projective Space of dimension 1 over Rational
Field
Defn: Defined on coordinates by sending (x : y) to
(x^2 - 2*y^2 : x*y)

sage: R.<t> = PolynomialRing(QQ)
sage: P.<x,y> = ProjectiveSpace(R,1)
sage: H = Hom(P,P)
sage: f = H([3/5*x^2,6*y^2])
sage: f.scale_by(5/3*t); f
Scheme endomorphism of Projective Space of dimension 1 over Univariate
Polynomial Ring in t over Rational Field
Defn: Defined on coordinates by sending (x : y) to
(t*x^2 : 10*t*y^2)

sage: P.<x,y,z> = ProjectiveSpace(GF(7),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.scale_by(x-y);f
Scheme endomorphism of Closed subscheme of Projective Space of dimension
2 over Finite Field of size 7 defined by:
x^2 - y^2
Defn: Defined on coordinates by sending (x : y : z) to
(x*y^2 - y^3 : x*y^2 - y^3 : x*z^2 - y*z^2)

sigma_invariants(n, formal=False, embedding=None, type='point')

Computes the values of the elementary symmetric polynomials of the n multiplier spectra of this dynamical system.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([512*x^5 - 378128*x^4*y + 76594292*x^3*y^2 - 4570550136*x^2*y^3\
- 2630045017*x*y^4 + 28193217129*y^5, 512*y^5])
sage: f.sigma_invariants(1)
doctest:warning
...
[19575526074450617/1048576, -9078122048145044298567432325/2147483648,
-2622661114909099878224381377917540931367/1099511627776,
-2622661107937102104196133701280271632423/549755813888,
338523204830161116503153209450763500631714178825448006778305/72057594037927936, 0]

wronskian_ideal()

Returns the ideal generated by the critical point locus.

This is the vanishing of the maximal minors of the Jacobian matrix. Not implemented for subvarieties.

OUTPUT: an ideal in the coordinate ring of the domain of this map.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<w> = NumberField(x^2+11)
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = End(P)
sage: f = H([x^2-w*y^2, w*y^2])
sage: f.wronskian_ideal()
Ideal ((4*w)*x*y) of Multivariate Polynomial Ring in x, y over Number
Field in w with defining polynomial x^2 + 11

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: P2.<u,v,t> = ProjectiveSpace(K,2)
sage: H = Hom(P,P2)
sage: f = H([x^2-2*y^2, y^2, x*y])
sage: f.wronskian_ideal()
Ideal (4*x*y, 2*x^2 + 4*y^2, -2*y^2) of Multivariate Polynomial Ring in
x, y over Rational Field

class sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_field(parent, polys, check=True)
all_rational_preimages(points)

Return the list of all rational preimages.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([16*x^2 - 29*y^2, 16*y^2])
sage: sorted(f.all_rational_preimages([P(-1,4)]))
doctest:warning
...
[(-7/4 : 1), (-5/4 : 1), (-3/4 : 1), (-1/4 : 1), (1/4 : 1), (3/4 : 1),
(5/4 : 1), (7/4 : 1)]

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = End(P)
sage: f = H([76*x^2 - 180*x*y + 45*y^2 + 14*x*z + 45*y*z - 90*z^2, 67*x^2 - 180*x*y - 157*x*z + 90*y*z, -90*z^2])
sage: sorted(f.all_rational_preimages([P(-9,-4,1)]))
[(-9 : -4 : 1), (0 : -1 : 1), (0 : 0 : 1), (0 : 1 : 1), (0 : 4 : 1), (1
: 0 : 1), (1 : 1 : 1), (1 : 2 : 1), (1 : 3 : 1)]


A non-periodic example

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2 + y^2, 2*x*y])
sage: sorted(f.all_rational_preimages([P(17,15)]))
[(1/3 : 1), (3/5 : 1), (5/3 : 1), (3 : 1)]


A number field example.:

sage: z = QQ['z'].0
sage: K.<w> = NumberField(z^3 + (z^2)/4 - (41/16)*z + 23/64);
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = End(P)
sage: f = H([16*x^2 - 29*y^2, 16*y^2])
sage: f.all_rational_preimages([P(16*w^2 - 29,16)])
[(-w^2 + 21/16 : 1),
(w : 1),
(w + 1/2 : 1),
(w^2 + w - 33/16 : 1),
(-w^2 - w + 25/16 : 1),
(w^2 - 21/16 : 1),
(-w^2 - w + 33/16 : 1),
(-w : 1),
(-w - 1/2 : 1),
(-w^2 + 29/16 : 1),
(w^2 - 29/16 : 1),
(w^2 + w - 25/16 : 1)]

sage: K.<w> = QuadraticField(3)
sage: P.<u,v> = ProjectiveSpace(K,1)
sage: H = End(P)
sage: f = H([u^2+v^2, v^2])
sage: f.all_rational_preimages(P(4))
[(-w : 1), (w : 1)]

conjugating_set(other)

Return the set of PGL element that conjugate this map to other.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(GF(7),1)
sage: H = End(P)
sage: D6 = H([y^2, x^2])
sage: D6.conjugating_set(D6)
doctest:warning
...
[
[1 0]  [0 1]  [0 2]  [4 0]  [2 0]  [0 4]
[0 1], [1 0], [1 0], [0 1], [0 1], [1 0]
]

connected_rational_component(P, n=0)

Return the component of all rational preimage and forward images.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<w> = NumberField(x^3+1/4*x^2-41/16*x+23/64)
sage: PS.<x,y> = ProjectiveSpace(1,K)
sage: H = End(PS)
sage: f = H([x^2 - 29/16*y^2, y^2])
sage: P = PS([w,1])
sage: f.connected_rational_component(P)
doctest:warning
...
[(w : 1),
(w^2 - 29/16 : 1),
(-w^2 - w + 25/16 : 1),
(w^2 + w - 25/16 : 1),
(-w : 1),
(-w^2 + 29/16 : 1),
(w + 1/2 : 1),
(-w - 1/2 : 1),
(-w^2 + 21/16 : 1),
(w^2 - 21/16 : 1),
(w^2 + w - 33/16 : 1),
(-w^2 - w + 33/16 : 1)]

indeterminacy_locus()

Return the indeterminacy locus of this map.

Only for rational maps on projective space defined over a field. The indeterminacy locus is the set of points in projective space at which all of the defining polynomials of the rational map simultaneously vanish.

OUTPUT:

• subscheme of the domain of the map. The empty subscheme is returned as the vanishing of the coordinate functions of the domain.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = End(P)
sage: f = H([x*z-y*z, x^2-y^2, z^2])
sage: f.indeterminacy_locus()
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x*z - y*z,
x^2 - y^2,
z^2

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = End(P)
sage: f = H([x^2, y^2, z^2])
sage: f.indeterminacy_locus()
Closed subscheme of Projective Space of dimension 2 over Rational Field
defined by:
x^2,
y^2,
z^2

sage: P1.<x,y,z> = ProjectiveSpace(RR,2)
sage: P2.<t,u,v,w> = ProjectiveSpace(RR,3)
sage: H = Hom(P1,P2)
sage: h = H([y^3*z^3, x^3*z^3, y^3*z^3, x^2*y^2*z^2])
sage: h.indeterminacy_locus()
Closed subscheme of Projective Space of dimension 2 over Real Field with
53 bits of precision defined by:
y^3*z^3,
x^3*z^3,
y^3*z^3,
x^2*y^2*z^2


If defining polynomials are not normalized, output scheme will not be normalized:

sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
sage: H=End(P)
sage: f=H([x*x^2,x*y^2,x*z^2])
sage: f.indeterminacy_locus()
Closed subscheme of Projective Space of dimension 2 over Rational Field
defined by:
x^3,
x*y^2,
x*z^2

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: X = P.subscheme(x-y)
sage: H = End(X)
sage: f = H([x^2-4*y^2, y^2-z^2, 4*z^2-x^2])
sage: Z = f.indeterminacy_locus(); Z
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
x - y,
x^2 - 4*y^2,
y^2 - z^2,
-x^2 + 4*z^2
sage: Z.dimension()
-1

indeterminacy_points(F=None)

Return the indeterminacy locus of this map defined over F.

Only for rational maps on projective space. Returns the set of points in projective space at which all of the defining polynomials of the rational map simultaneously vanish.

INPUT:

• F - a field (optional).

OUTPUT:

• indeterminacy points of the map defined over F, provided the indeterminacy scheme is 0-dimensional.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = End(P)
sage: f = H([x*z-y*z, x^2-y^2, z^2])
sage: f.indeterminacy_points()
[(-1 : 1 : 0), (1 : 1 : 0)]

sage: P1.<x,y,z> = ProjectiveSpace(RR,2)
sage: P2.<t,u,v,w> = ProjectiveSpace(RR,3)
sage: H = Hom(P1,P2)
sage: h = H([x+y, y, z+y, y])
sage: h.indeterminacy_points()
[]
sage: g = H([y^3*z^3, x^3*z^3, y^3*z^3, x^2*y^2*z^2])
sage: g.indeterminacy_points()
Traceback (most recent call last):
...
ValueError: indeterminacy scheme is not dimension 0

sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: H = End(P)
sage: f = H([x^2+y^2, x*z, x^2+y^2])
sage: f.indeterminacy_points()
[(0 : 0 : 1)]
sage: R.<t> = QQ[]
sage: K.<a> = NumberField(t^2+1)
sage: f.indeterminacy_points(F=K)
[(-a : 1 : 0), (0 : 0 : 1), (a : 1 : 0)]
sage: set_verbose(None)
sage: f.indeterminacy_points(F=QQbar)
[(-1*I : 1 : 0), (0 : 0 : 1), (1*I : 1 : 0)]

sage: set_verbose(None)
sage: K.<t>=FunctionField(QQ)
sage: P.<x,y,z>=ProjectiveSpace(K,2)
sage: H=End(P)
sage: f=H([x^2-t^2*y^2,y^2-z^2,x^2-t^2*z^2])
sage: f.indeterminacy_points()
[(-t : -1 : 1), (-t : 1 : 1), (t : -1 : 1), (t : 1 : 1)]

sage: set_verbose(None)
sage: P.<x,y,z>=ProjectiveSpace(Qp(3),2)
sage: H=End(P)
sage: f=H([x^2-7*y^2,y^2-z^2,x^2-7*z^2])
sage: f.indeterminacy_points()
[(2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 2*3^6 + 3^8 + 3^9 + 2*3^11 + 3^15 +
2*3^16 + 3^18 + O(3^20) : 1 + O(3^20) : 1 + O(3^20)),
(2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 2*3^6 + 3^8 + 3^9 + 2*3^11 + 3^15 +
2*3^16 + 3^18 + O(3^20) : 2 + 2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 +
2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11 + 2*3^12 + 2*3^13 +
2*3^14 + 2*3^15 + 2*3^16 + 2*3^17 + 2*3^18 + 2*3^19 + O(3^20) : 1 +
O(3^20)),
(1 + 3 + 3^2 + 2*3^4 + 2*3^7 + 3^8 + 3^9 + 2*3^10 + 2*3^12 + 2*3^13 +
2*3^14 + 3^15 + 2*3^17 + 3^18 + 2*3^19 + O(3^20) : 1 + O(3^20) : 1 +
O(3^20)),
(1 + 3 + 3^2 + 2*3^4 + 2*3^7 + 3^8 + 3^9 + 2*3^10 + 2*3^12 + 2*3^13 +
2*3^14 + 3^15 + 2*3^17 + 3^18 + 2*3^19 + O(3^20) : 2 + 2*3 + 2*3^2 +
2*3^3 + 2*3^4 + 2*3^5 + 2*3^6 + 2*3^7 + 2*3^8 + 2*3^9 + 2*3^10 + 2*3^11
+ 2*3^12 + 2*3^13 + 2*3^14 + 2*3^15 + 2*3^16 + 2*3^17 + 2*3^18 + 2*3^19
+ O(3^20) : 1 + O(3^20))]

is_conjugate(other)

Return if this map and other are conjugate.

EXAMPLES:

sage: K.<w> = CyclotomicField(3)
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = End(P)
sage: D8 = H([y^2, x^2])
sage: D8.is_conjugate(D8)
doctest:warning
...
True

is_polynomial()

Return if this map is a polynomial.

EXAMPLES:

sage: R.<x> = QQ[]
sage: P.<x,y> = ProjectiveSpace(K, 1)
sage: H = End(P)
sage: f = H([x**2 + 2*x*y - 5*y**2, 2*x*y])
sage: f.is_polynomial()
doctest:warning
...
False

lift_to_rational_periodic(points_modp, B=None)

Return the rational lift of the modp periodic points.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2 - y^2, y^2])
sage: f.lift_to_rational_periodic([[P(0,1).change_ring(GF(7)), 4]])
doctest:warning
...
[[(0 : 1), 2]]

normal_form(return_conjugation=False)

Return a normal form conjugate to this map.

EXAMPLES:

sage: R.<x> = QQ[]
sage: K.<w> = NumberField(x^2 - 5)
sage: P.<x,y> = ProjectiveSpace(K,1)
sage: H = End(P)
sage: f = H([x^2 + w*x*y, y^2])
sage: g,m,psi = f.normal_form(return_conjugation = True);m
doctest:warning
...
[     1 -1/2*w]
[     0      1]

rational_periodic_points(**kwds)

Return the list of all rational periodic points.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = End(P)
sage: f = H([x^2-3/4*y^2, y^2])
sage: sorted(f.rational_periodic_points(prime_bound=20, lifting_prime=7)) # long time
doctest:warning
...
[(-1/2 : 1), (1 : 0), (3/2 : 1)]

rational_preimages(Q, k=1)

Determine all of the rational $$k$$-th preimages of Q by this map.

Given a rational point Q in the domain of this map, return all the rational points P in the domain with $$f^k(P)==Q$$. In other words, the set of $$k$$-th preimages of Q. The map must be defined over a number field and be an endomorphism for $$k > 1$$.

If Q is a subscheme, then return the subscheme that maps to Q by this map. In particular, $$f^{-k}(V(h_1,\ldots,h_t)) = V(h_1 \circ f^k, \ldots, h_t \circ f^k)$$.

INPUT:

• Q - a rational point or subscheme in the domain of this map.
• k - positive integer.

OUTPUT:

• a list of rational points or a subscheme in the domain of this map.

Examples:

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([16*x^2 - 29*y^2, 16*y^2])
sage: f.rational_preimages(P(-1, 4))
[(-5/4 : 1), (5/4 : 1)]

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: H = End(P)
sage: f = H([76*x^2 - 180*x*y + 45*y^2 + 14*x*z + 45*y*z\
- 90*z^2, 67*x^2 - 180*x*y - 157*x*z + 90*y*z, -90*z^2])
sage: f.rational_preimages(P(-9, -4, 1))
[(0 : 4 : 1)]


A non-periodic example

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([x^2 + y^2, 2*x*y])
sage: f.rational_preimages(P(17, 15))
[(3/5 : 1), (5/3 : 1)]

sage: P.<x,y,z,w> = ProjectiveSpace(QQ, 3)
sage: H = End(P)
sage: f = H([x^2 - 2*y*w - 3*w^2, -2*x^2 + y^2 - 2*x*z\
+ 4*y*w + 3*w^2, x^2 - y^2 + 2*x*z + z^2 - 2*y*w - w^2, w^2])
sage: f.rational_preimages(P(0, -1, 0, 1))
[]

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([x^2 + y^2, 2*x*y])
sage: f.rational_preimages([CC.0, 1])
Traceback (most recent call last):
...
TypeError: point must be in codomain of self


A number field example

sage: z = QQ['z'].0
sage: K.<a> = NumberField(z^2 - 2);
sage: P.<x,y> = ProjectiveSpace(K, 1)
sage: H = End(P)
sage: f = H([x^2 + y^2, y^2])
sage: f.rational_preimages(P(3, 1))
[(-a : 1), (a : 1)]

sage: z = QQ['z'].0
sage: K.<a> = NumberField(z^2 - 2);
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: X = P.subscheme([x^2 - z^2])
sage: H = End(X)
sage: f= H([x^2 - z^2, a*y^2, z^2 - x^2])
sage: f.rational_preimages(X([1, 2, -1]))
[]

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
sage: X = P.subscheme([x^2 - z^2])
sage: H = End(X)
sage: f= H([x^2-z^2, y^2, z^2-x^2])
sage: f.rational_preimages(X([0, 1, 0]))
Traceback (most recent call last):
...
NotImplementedError: subschemes as preimages not implemented

sage: P.<x, y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([x^2-y^2, y^2])
sage: f.rational_preimages(P.subscheme([x]))
Closed subscheme of Projective Space of dimension 1 over Rational Field
defined by:
x^2 - y^2

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: H = End(P)
sage: f = H([x^2 - 29/16*y^2, y^2])
sage: f.rational_preimages(P(5/4, 1), k=4)
[(-3/4 : 1), (3/4 : 1), (-7/4 : 1), (7/4 : 1)]

sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: P2.<u,v,w> = ProjectiveSpace(QQ, 2)
sage: H = Hom(P, P2)
sage: f = H([x^2, y^2, x^2-y^2])
sage: f.rational_preimages(P2(1, 1, 0))
[(-1 : 1), (1 : 1)]

rational_preperiodic_graph(**kwds)

Return the digraph of rational preperiodic points.

EXAMPLES:

sage: PS.<x,y> = ProjectiveSpace(1,QQ)
sage: H = End(PS)
sage: f = H([7*x^2 - 28*y^2, 24*x*y])
sage: f.rational_preperiodic_graph()
doctest:warning
...
Looped digraph on 12 vertices

rational_preperiodic_points(**kwds)

Return the list of all rational preperiodic points.

EXAMPLES:

sage: PS.<x,y> = ProjectiveSpace(1,QQ)
sage: H = End(PS)
sage: f = H([x^2 -y^2, 3*x*y])
sage: sorted(f.rational_preperiodic_points())
doctest:warning
...
[(-2 : 1), (-1 : 1), (-1/2 : 1), (0 : 1), (1/2 : 1), (1 : 0), (1 : 1),
(2 : 1)]

class sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field(parent, polys, check=True)
automorphism_group(**kwds)

return the automorphism group of this map.

EXAMPLES:

sage: R.<x,y> = ProjectiveSpace(GF(7^3,'t'),1)
sage: H = End(R)
sage: f = H([x^2-y^2, x*y])
sage: f.automorphism_group()
doctest:warning
...
[
[1 0]  [6 0]
[0 1], [0 1]
]

cyclegraph()

Return the digraph of the map.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(GF(13),1)
sage: H = End(P)
sage: f = H([x^2-y^2, y^2])
sage: f.cyclegraph()
doctest:warning
...
Looped digraph on 14 vertices

orbit_structure(P)

Return the tail and period of the point.

EXAMPLES:

sage: P.<x,y,z> = ProjectiveSpace(GF(5),2)
sage: H = End(P)
sage: f = H([x^2 + y^2,y^2, z^2 + y*z])
sage: f.orbit_structure(P(2,1,2))
doctest:warning
...
[0, 6]

possible_periods(return_points=False)

Return the possible periods of a periodic point of this map.

EXAMPLES:

sage: P.<x,y> = ProjectiveSpace(GF(23),1)
sage: H = End(P)
sage: f = H([x^2-2*y^2, y^2])
sage: f.possible_periods()
doctest:warning
...
[1, 5, 11, 22, 110]