# -*- coding: utf-8 -*-
r"""
Dynamical systems on projective schemes

A dynamical system of projective schemes determined by homogeneous
polynomials functions that define what the morphism does on points
in the ambient projective space.

The main constructor functions are given by :class:`DynamicalSystem` and
:class:`DynamicalSystem_projective`. The constructors function can take either
polynomials or a morphism 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.

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:

- 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

- Ben Hutz (2017-7): relocate code and create class

"""

#*****************************************************************************
#       Copyright (C) 2011 Volker Braun <vbraun.name@gmail.com>
#       Copyright (C) 2006 David Kohel <kohel@maths.usyd.edu.au>
#       Copyright (C) 2006 William Stein <wstein@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  http://www.gnu.org/licenses/
#*****************************************************************************
from __future__ import print_function
from __future__ import absolute_import

from sage.arith.misc import is_prime
from sage.categories.fields import Fields
from sage.categories.function_fields import FunctionFields
from sage.categories.number_fields import NumberFields
from sage.categories.homset import End
from sage.dynamics.arithmetic_dynamics.generic_ds import DynamicalSystem
from sage.functions.all import sqrt
from sage.functions.other import ceil
from sage.libs.pari.all import PariError
from sage.matrix.constructor import matrix, identity_matrix
from sage.misc.cachefunc import cached_method
from sage.misc.classcall_metaclass import typecall
from sage.misc.mrange import xmrange
from sage.modules.free_module_element import vector
from sage.rings.all import Integer, CIF
from sage.arith.all import gcd, lcm, next_prime, binomial, primes, moebius
from sage.categories.finite_fields import FiniteFields
from sage.rings.finite_rings.finite_field_constructor import (is_FiniteField, GF,
                                                              is_PrimeFiniteField)
from sage.rings.finite_rings.integer_mod_ring import Zmod
from sage.rings.fraction_field import (FractionField, is_FractionField)
from sage.rings.fraction_field_element import is_FractionFieldElement, FractionFieldElement
from sage.rings.integer_ring import ZZ
from sage.rings.morphism import RingHomomorphism_im_gens
from sage.rings.number_field.number_field_ideal import NumberFieldFractionalIdeal
from sage.rings.padics.all import Qp
from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.polynomial.polynomial_ring import is_PolynomialRing
from sage.rings.qqbar import QQbar
from sage.rings.quotient_ring import QuotientRing_generic
from sage.rings.rational_field import QQ
from sage.rings.real_double import RDF
from sage.rings.real_mpfr import (RealField, is_RealField)
from sage.schemes.generic.morphism import SchemeMorphism_polynomial
from sage.schemes.projective.projective_subscheme import AlgebraicScheme_subscheme_projective
from sage.schemes.projective.projective_morphism import (
    SchemeMorphism_polynomial_projective_space,
    SchemeMorphism_polynomial_projective_space_field,
    SchemeMorphism_polynomial_projective_space_finite_field)
from sage.schemes.projective.projective_space import (ProjectiveSpace,
                                                      is_ProjectiveSpace)
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
from sage.symbolic.constants import e
from copy import copy
from sage.parallel.ncpus import ncpus
from sage.parallel.use_fork import p_iter_fork
from sage.dynamics.arithmetic_dynamics.projective_ds_helper import _fast_possible_periods
from sage.sets.set import Set
from sage.combinat.permutation import Arrangements
from sage.combinat.subset import Subsets
from sage.symbolic.ring import SR

class DynamicalSystem_projective(SchemeMorphism_polynomial_projective_space,
                                      DynamicalSystem):
    r"""A dynamical system of projective schemes determined by homogeneous
    polynomials that define what the morphism does on points in the
    ambient projective space.

    .. WARNING::

        You should not create objects of this class directly because
        no type or consistency checking is performed. The preferred
        method to construct such dynamical systems is to use
        :func:`~sage.dynamics.arithmetic_dynamics.generic_ds.DynamicalSystem_projective`
        function

    INPUT:

    - ``morphism_or_polys`` -- a SchemeMorphism, a polynomial, a
      rational function, or a list or tuple of homogeneous polynomials.

    - ``domain`` -- optional projective space or projective subscheme.

    - ``names`` -- optional tuple of strings to be used as coordinate
      names for a projective space that is constructed; defaults to ``'X','Y'``.

      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 base ring of
        the first element of ``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``.

    OUTPUT: :class:`DynamicalSystem_projectve`.

    EXAMPLES::

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

    We can define dynamical systems on `P^1` by giving a polynomial or
    rational function::

        sage: R.<t> = QQ[]
        sage: DynamicalSystem_projective(t^2 - 3)
        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 : Y^2)
        sage: DynamicalSystem_projective(1/t^2)
        Dynamical System of Projective Space of dimension 1 over Rational Field
          Defn: Defined on coordinates by sending (X : Y) to
                (Y^2 : X^2)

    ::

        sage: R.<x> = PolynomialRing(QQ,1)
        sage: DynamicalSystem_projective(x^2, names=['a','b'])
        Dynamical System of Projective Space of dimension 1 over Rational Field
          Defn: Defined on coordinates by sending (a : b) to
                (a^2 : b^2)

    Symbolic Ring elements are not allows::

        sage: x,y = var('x,y')
        sage: DynamicalSystem_projective([x^2,y^2])
        Traceback (most recent call last):
        ...
        ValueError: [x^2, y^2] must be elements of a polynomial ring

    ::

        sage: R.<x> = PolynomialRing(QQ,1)
        sage: DynamicalSystem_projective(x^2)
        Dynamical System of Projective Space of dimension 1 over Rational Field
          Defn: Defined on coordinates by sending (X : Y) to
                (X^2 : Y^2)

    ::

        sage: R.<t> = PolynomialRing(QQ)
        sage: P.<x,y,z> = ProjectiveSpace(R, 2)
        sage: X = P.subscheme([x])
        sage: DynamicalSystem_projective([x^2, t*y^2, x*z], domain=X)
        Dynamical System of Closed subscheme of Projective Space of dimension
        2 over Univariate Polynomial Ring in t over Rational Field defined by:
          x
          Defn: Defined on coordinates by sending (x : y : z) to
                (x^2 : t*y^2 : x*z)

    When elements of the quotient ring are used, they are reduced::

        sage: P.<x,y,z> = ProjectiveSpace(CC, 2)
        sage: X = P.subscheme([x-y])
        sage: u,v,w = X.coordinate_ring().gens()
        sage: DynamicalSystem_projective([u^2, v^2, w*u], domain=X)
        Dynamical System of Closed subscheme of Projective Space of dimension
        2 over Complex Field with 53 bits of precision defined by:
          x - y
          Defn: Defined on coordinates by sending (x : y : z) to
                (y^2 : y^2 : y*z)

    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: f = DynamicalSystem_projective([(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: f = DynamicalSystem_projective([(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

    ::

        sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ)
        sage: DynamicalSystem_projective([x^2*u, y^2*w, z^2*u, w^2, u^2], domain=T)
        Dynamical System of Product of projective spaces P^2 x P^1 over Rational Field
          Defn: Defined by sending (x : y : z , w : u) to
                (x^2*u : y^2*w : z^2*u , w^2 : u^2).
    """

    @staticmethod
    def __classcall_private__(cls, morphism_or_polys, domain=None, names=None):
        r"""
        Return the appropriate dynamical system on a projective scheme.

        TESTS::

            sage: R.<x,y> = QQ[]
            sage: P1 = ProjectiveSpace(R)
            sage: f = DynamicalSystem_projective([x-y, x*y])
            Traceback (most recent call last):
            ...
            ValueError: polys (=[x - y, x*y]) must be of the same degree
            sage: DynamicalSystem_projective([x-1, x*y+x])
            Traceback (most recent call last):
            ...
            ValueError: polys (=[x - 1, x*y + x]) must be homogeneous

        ::

            sage: DynamicalSystem_projective([exp(x),exp(y)])
            Traceback (most recent call last):
            ...
            ValueError: [e^x, e^y] must be elements of a polynomial ring

        ::

            sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2, 1], QQ)
            sage: DynamicalSystem_projective([x^2*u, y^2*w, z^2*u, w^2, u*z], domain=T)
            Traceback (most recent call last):
            ...
            TypeError: polys (=[x^2*u, y^2*w, z^2*u, w^2, z*u]) must be
            multi-homogeneous of the same degrees (by component)

        ::

            sage: A.<x,y> = AffineSpace(ZZ, 2)
            sage: DynamicalSystem_projective([x^2,y^2], A)
            Traceback (most recent call last):
            ...
            ValueError: "domain" must be a projective scheme
            sage: H = End(A)
            sage: f = H([x,y])
            sage: DynamicalSystem_projective(f)
            Traceback (most recent call last):
            ...
            ValueError: "domain" must be a projective scheme

        ::

            sage: R.<x> = PolynomialRing(QQ,1)
            sage: DynamicalSystem_projective(x^2, names='t')
            Traceback (most recent call last):
            ...
            ValueError: specify 2 variable names

        ::

            sage: P1.<x,y> = ProjectiveSpace(QQ,1)
            sage: DynamicalSystem_projective([y, x, y], domain=P1)
            Traceback (most recent call last):
            ...
            ValueError: polys (=[y, x, y]) do not define a rational endomorphism of the domain

        ::

            sage: A.<x,y> = AffineSpace(QQ,2)
            sage: DynamicalSystem_projective([y,x], domain=A)
            Traceback (most recent call last):
            ...
            ValueError: "domain" must be a projective scheme

        ::

            sage: R.<x> = QQ[]
            sage: DynamicalSystem([x^2])
            Traceback (most recent call last):
            ...
            ValueError: list/tuple must have at least 2 polynomials
        """
        from sage.dynamics.arithmetic_dynamics.product_projective_ds import DynamicalSystem_product_projective

        if isinstance(morphism_or_polys, SchemeMorphism_polynomial):
            R = morphism_or_polys.base_ring()
            domain = morphism_or_polys.domain()
            polys = list(morphism_or_polys)
            if domain != morphism_or_polys.codomain():
                raise ValueError('domain and codomain do not agree')
            if not is_ProjectiveSpace(domain) and not isinstance(domain, AlgebraicScheme_subscheme_projective):
                raise ValueError('"domain" must be a projective scheme')
            if R not in Fields():
                return typecall(cls, polys, domain)
            if is_FiniteField(R):
                return DynamicalSystem_projective_finite_field(polys, domain)
            return DynamicalSystem_projective_field(polys, domain)

        if isinstance(morphism_or_polys, (list, tuple)):
            polys = list(morphism_or_polys)
            if len(polys) == 1:
                raise ValueError("list/tuple must have at least 2 polynomials")
            test = lambda x: is_PolynomialRing(x) or is_MPolynomialRing(x)
            if not all(test(poly.parent()) for poly in polys):
                try:
                    polys = [poly.lift() for poly in polys]
                except:
                    raise ValueError('{} must be elements of a polynomial ring'.format(morphism_or_polys))
        else:
            # homogenize!
            f = morphism_or_polys
            aff_CR = f.parent()
            if (not is_PolynomialRing(aff_CR) and not is_FractionField(aff_CR)
                and not (is_MPolynomialRing(aff_CR) and aff_CR.ngens() == 1)):
                msg = '{} is not a single variable polynomial or rational function'
                raise ValueError(msg.format(f))
            if is_FractionField(aff_CR):
                polys = [f.numerator(),f.denominator()]
            else:
                polys = [f, aff_CR(1)]
            d = max(poly.degree() for poly in polys)
            if names is None:
                names = ('X','Y')
            elif len(names) != 2:
                raise ValueError('specify 2 variable names')
            proj_CR = PolynomialRing(aff_CR.base_ring(), names=names)
            X,Y = proj_CR.gens()
            polys = [proj_CR(Y**d * poly(X/Y)) for poly in polys]
        if domain is None:
            f = polys[0]
            proj_CR = f.parent()
            domain = ProjectiveSpace(proj_CR)
        R = domain.base_ring()
        if R is SR:
            raise TypeError("Symbolic Ring cannot be the base ring")

        if len(polys) != domain.ambient_space().coordinate_ring().ngens():
            msg = 'polys (={}) do not define a rational endomorphism of the domain'
            raise ValueError(msg.format(polys))

        if is_ProductProjectiveSpaces(domain):
            splitpolys = domain._factors(polys)
            for split_poly in splitpolys:
                split_d = domain._degree(split_poly[0])
                if not all(split_d == domain._degree(f) for f in split_poly):
                    msg = 'polys (={}) must be multi-homogeneous of the same degrees (by component)'
                    raise TypeError(msg.format(polys))
            return DynamicalSystem_product_projective(polys, domain)

        # Now polys define an endomorphism of a scheme in P^n
        if not all(poly.is_homogeneous() for poly in polys):
            msg = 'polys (={}) must be homogeneous'
            raise ValueError(msg.format(polys))
        d = polys[0].degree()
        if not all(poly.degree() == d for poly in polys):
            msg = 'polys (={}) must be of the same degree'
            raise ValueError(msg.format(polys))

        if not is_ProjectiveSpace(domain) and not isinstance(domain, AlgebraicScheme_subscheme_projective):
            raise ValueError('"domain" must be a projective scheme')
        if R not in Fields():
            return typecall(cls, polys, domain)
        if is_FiniteField(R):
                return DynamicalSystem_projective_finite_field(polys, domain)
        return DynamicalSystem_projective_field(polys, domain)

    def __init__(self, polys, domain):
        r"""
        The Python constructor.

        See :class:`DynamicalSystem` for details.

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: DynamicalSystem_projective([3/5*x^2, y^2], domain=P)
            Dynamical System of Projective Space of dimension 1 over Rational Field
              Defn: Defined on coordinates by sending (x : y) to
                    (3/5*x^2 : y^2)
        """
        # Next attribute needed for _fast_eval and _fastpolys
        self._is_prime_finite_field = is_PrimeFiniteField(polys[0].base_ring())
        DynamicalSystem.__init__(self,polys,domain)

    def __copy__(self):
        r"""
        Return a copy of this dynamical system.

        OUTPUT: :class:`DynamicalSystem_projective`

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([3/5*x^2,6*y^2])
            sage: g = copy(f)
            sage: f == g
            True
            sage: f is g
            False
        """
        return DynamicalSystem_projective(self._polys, self.domain())

    def dehomogenize(self, n):
        r"""
        Return 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:

        :class:`DynamicalSystem_affine` given by dehomogenizing the
        source and target of `self` with respect to the given indices.

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2])
            sage: f.dehomogenize(0)
            Dynamical System of Affine Space of dimension 1 over Integer Ring
              Defn: Defined on coordinates by sending (x) to
                    (x^2/(x^2 + 1))
        """
        F = self.as_scheme_morphism().dehomogenize(n)
        return F.as_dynamical_system()

    def dynatomic_polynomial(self, period):
        r"""
        For a dynamical system of `\mathbb{P}^1` compute the dynatomic
        polynomial.

        The dynatomic polynomial is the analog of the cyclotomic
        polynomial and its roots are the points of formal period `period`.
        If possible the division is done in the coordinate ring of this
        map and a polynomial is returned. In rings where that is not
        possible, a :class:`FractionField` element will be returned.
        In certain cases, when the conversion back to a polynomial fails,
        a :class:`SymbolRing` element will be returned.

        ALGORITHM:

        For a positive integer `n`, let `[F_n,G_n]` be the coordinates of the `nth`
        iterate of `f`. Then construct

        .. MATH::

            \Phi^{\ast}_n(f)(x,y) = \sum_{d \mid n}
                (yF_d(x,y) - xG_d(x,y))^{\mu(n/d)},

        where `\mu` is the Möbius function.

        For a pair `[m,n]`, let `f^m = [F_m,G_m]`. Compute

        .. MATH::

            \Phi^{\ast}_{m,n}(f)(x,y) = \Phi^{\ast}_n(f)(F_m,G_m) /
                \Phi^{\ast}_n(f)(F_{m-1},G_{m-1})

        REFERENCES:

        - [Hutz2015]_
        - [MoPa1994]_

        INPUT:

        - ``period`` -- a positive integer or a list/tuple `[m,n]` where
          `m` is the preperiod and `n` is the period

        OUTPUT:

        If possible, a two variable polynomial in the coordinate ring
        of this map. Otherwise a fraction field element of the coordinate
        ring of this map. Or, a :class:`SymbolicRing` element.

        .. TODO::

            - Do the division when the base ring is `p`-adic so that
              the output is a polynomial.

            - Convert back to a polynomial when the base ring is a
              function field (not over `\QQ` or `F_p`).

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, x*y])
            sage: f.dynatomic_polynomial(4)
            2*x^12 + 18*x^10*y^2 + 57*x^8*y^4 + 79*x^6*y^6 + 48*x^4*y^8 + 12*x^2*y^10 + y^12

        ::

            sage: P.<x,y> = ProjectiveSpace(CC,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, 3*x*y])
            sage: f.dynatomic_polynomial(3)
            13.0000000000000*x^6 + 117.000000000000*x^4*y^2 +
            78.0000000000000*x^2*y^4 + y^6

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 - 10/9*y^2, y^2])
            sage: f.dynatomic_polynomial([2,1])
            x^4*y^2 - 11/9*x^2*y^4 - 80/81*y^6

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 - 29/16*y^2, y^2])
            sage: f.dynatomic_polynomial([2,3])
            x^12 - 95/8*x^10*y^2 + 13799/256*x^8*y^4 - 119953/1024*x^6*y^6 +
            8198847/65536*x^4*y^8 - 31492431/524288*x^2*y^10 +
            172692729/16777216*y^12

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2 - y^2, y^2])
            sage: f.dynatomic_polynomial([1,2])
            x^2 - x*y

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^3 - y^3, 3*x*y^2])
            sage: f.dynatomic_polynomial([0,4])==f.dynatomic_polynomial(4)
            True

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x^2 + y^2, x*y, z^2])
            sage: f.dynatomic_polynomial(2)
            Traceback (most recent call last):
            ...
            TypeError: does not make sense in dimension >1

        ::

            sage: P.<x,y> = ProjectiveSpace(Qp(5),1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, y^2])
            sage: f.dynatomic_polynomial(2)
            (x^4*y + (2 + O(5^20))*x^2*y^3 - x*y^4 + (2 + O(5^20))*y^5)/(x^2*y -
            x*y^2 + y^3)

        ::

            sage: L.<t> = PolynomialRing(QQ)
            sage: P.<x,y> = ProjectiveSpace(L,1)
            sage: f = DynamicalSystem_projective([x^2 + t*y^2, y^2])
            sage: f.dynatomic_polynomial(2)
            x^2 + x*y + (t + 1)*y^2

        ::

            sage: K.<c> = PolynomialRing(ZZ)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2 + c*y^2, y^2])
            sage: f.dynatomic_polynomial([1, 2])
            x^2 - x*y + (c + 1)*y^2

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, y^2])
            sage: f.dynatomic_polynomial(2)
            x^2 + x*y + 2*y^2
            sage: R.<X> = PolynomialRing(QQ)
            sage: K.<c> = NumberField(X^2 + X + 2)
            sage: PP = P.change_ring(K)
            sage: ff = f.change_ring(K)
            sage: p = PP((c, 1))
            sage: ff(ff(p)) == p
            True

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, x*y])
            sage: f.dynatomic_polynomial([2, 2])
            x^4 + 4*x^2*y^2 + y^4
            sage: R.<X> = PolynomialRing(QQ)
            sage: K.<c> = NumberField(X^4 + 4*X^2 + 1)
            sage: PP = P.change_ring(K)
            sage: ff = f.change_ring(K)
            sage: p = PP((c, 1))
            sage: ff.nth_iterate(p, 4) == ff.nth_iterate(p, 2)
            True

        ::

            sage: P.<x,y> = ProjectiveSpace(CC, 1)
            sage: f = DynamicalSystem_projective([x^2 - CC.0/3*y^2, y^2])
            sage: f.dynatomic_polynomial(2)
            (x^4*y + (-0.666666666666667*I)*x^2*y^3 - x*y^4 + (-0.111111111111111 - 0.333333333333333*I)*y^5)/(x^2*y - x*y^2 + (-0.333333333333333*I)*y^3)

        ::

            sage: P.<x,y> = ProjectiveSpace(CC, 1)
            sage: f = DynamicalSystem_projective([x^2-CC.0/5*y^2, y^2])
            sage: f.dynatomic_polynomial(2)
            x^2 + x*y + (1.00000000000000 - 0.200000000000000*I)*y^2

        ::

            sage: L.<t> = PolynomialRing(QuadraticField(2).maximal_order())
            sage: P.<x, y> = ProjectiveSpace(L.fraction_field() , 1)
            sage: f = DynamicalSystem_projective([x^2 + (t^2 + 1)*y^2 , y^2])
            sage: f.dynatomic_polynomial(2)
            x^2 + x*y + (t^2 + 2)*y^2

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
            sage: f = DynamicalSystem_projective([x^2 - 5*y^2, y^2])
            sage: f.dynatomic_polynomial([3,0 ])
            0

        TESTS:

        We check that the dynatomic polynomial has the right
        parent (see :trac:`18409`)::

            sage: P.<x,y> = ProjectiveSpace(QQbar,1)
            sage: f = DynamicalSystem_projective([x^2 - 1/3*y^2, y^2])
            sage: f.dynatomic_polynomial(2).parent()
            Multivariate Polynomial Ring in x, y over Algebraic Field

        ::

            sage: T.<v> = QuadraticField(33)
            sage: S.<t> = PolynomialRing(T)
            sage: P.<x,y> = ProjectiveSpace(FractionField(S),1)
            sage: f = DynamicalSystem_projective([t*x^2 - 1/t*y^2, y^2])
            sage: f.dynatomic_polynomial([1, 2]).parent()
            Multivariate Polynomial Ring in x, y over Fraction Field of Univariate Polynomial
            Ring in t over Number Field in v with defining polynomial x^2 - 33

        ::

            sage: P.<x, y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^3 - y^3*2, y^3])
            sage: f.dynatomic_polynomial(1).parent()
            Multivariate Polynomial Ring in x, y over Rational Field

        ::

            sage: R.<c> = QQ[]
            sage: P.<x,y> = ProjectiveSpace(R,1)
            sage: f = DynamicalSystem_projective([x^2 + c*y^2, y^2])
            sage: f.dynatomic_polynomial([1,2]).parent()
            Multivariate Polynomial Ring in x, y over Univariate
            Polynomial Ring in c over Rational Field

        ::

            sage: R.<c> = QQ[]
            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, (1)*y^2 + (1)*x*y])
            sage: f.dynatomic_polynomial([1,2]).parent()
            Multivariate Polynomial Ring in x, y over Integer Ring

        ::

            sage: P.<x, y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, y^2])
            sage: f.dynatomic_polynomial(0)
            0
            sage: f.dynatomic_polynomial([0,0])
            0
            sage: f.dynatomic_polynomial(-1)
            Traceback (most recent call last):
            ...
            TypeError: period must be a positive integer

        ::

            sage: R.<c> = QQ[]
            sage: P.<x,y> = ProjectiveSpace(R,1)
            sage: f = DynamicalSystem_projective([x^2 + c*y^2,y^2])
            sage: f.dynatomic_polynomial([1,2]).parent()
            Multivariate Polynomial Ring in x, y over Univariate Polynomial Ring in
            c over Rational Field

        Some rings still return :class:`SymoblicRing` elements::

            sage: S.<t> = FunctionField(CC)
            sage: P.<x,y> = ProjectiveSpace(S,1)
            sage: f = DynamicalSystem_projective([t*x^2-1*y^2, t*y^2])
            sage: f.dynatomic_polynomial([1, 2]).parent()
            Symbolic Ring

        ::

            sage: R.<x,y> = PolynomialRing(QQ)
            sage: S = R.quo(R.ideal(y^2-x+1))
            sage: P.<u,v> = ProjectiveSpace(FractionField(S),1)
            sage: f = DynamicalSystem_projective([u^2 + S(x^2)*v^2, v^2])
            sage: dyn = f.dynatomic_polynomial([1,1]); dyn
            v^3*xbar^2 + u^2*v + u*v^2
            sage: dyn.parent()
            Symbolic Ring
        """
        if self.domain().ngens() > 2:
            raise TypeError("does not make sense in dimension >1")
        if not isinstance(period, (list, tuple)):
            period = [0, period]
        x = self.domain().gen(0)
        y = self.domain().gen(1)
        f0, f1 = F0, F1 = self._polys
        PHI = self.base_ring().one()
        m = period[0]
        n = int(period[1])
        if n < 0:
            raise TypeError("period must be a positive integer")
        if n == 0:
            return self[0].parent().zero()
        if m == 0 and n == 1:
            return y*F0 - x*F1
        for d in range(1, n):
            if n % d == 0:
                PHI = PHI * ((y*F0 - x*F1)**moebius(n//d))
            F0, F1 = f0(F0, F1), f1(F0, F1)
        PHI = PHI * (y*F0 - x*F1)
        if m != 0:
            fm = self.nth_iterate_map(m)
            fm1 = self.nth_iterate_map(m - 1)
        try:
            QR = PHI.numerator().quo_rem(PHI.denominator())
            if not QR[1]:
                PHI = QR[0]
            if m != 0:
                PHI = PHI(fm._polys)/(PHI(fm1._polys))
                QR = PHI.numerator().quo_rem(PHI.denominator())
                if QR[1] == 0:
                    PHI = QR[0]
            return PHI
        except (TypeError, NotImplementedError): # something Singular can't handle
            if m != 0:
                PHI = PHI(fm._polys) / PHI(fm1._polys)
        #even when the ring can be passed to singular in quo_rem,
        #it can't always do the division, so we call Maxima
        from sage.rings.padics.generic_nodes import is_pAdicField, is_pAdicRing
        if period != [0,1]: #period==[0,1] we don't need to do any division
            BR = self.domain().base_ring().base_ring()
            if not (is_pAdicRing(BR) or is_pAdicField(BR)):
                try:
                    QR2 = PHI.numerator()._maxima_().divide(PHI.denominator())
                    if not QR2[1].sage():
                        # do it again to divide out by denominators of coefficients
                        PHI = QR2[0].sage()
                        PHI = PHI.numerator()._maxima_().divide(PHI.denominator())[0].sage()
                    if not is_FractionFieldElement(PHI):
                        from sage.symbolic.expression_conversions import polynomial
                        PHI = polynomial(PHI, ring=self.coordinate_ring())
                except (TypeError, NotImplementedError): #something Maxima, or the conversion, can't handle
                    pass
        return PHI

    def nth_iterate_map(self, n, normalize=False):
        r"""
        Return the ``n``-th iterate of this dynamical system.

        ALGORITHM:

        Uses a form of successive squaring to reducing computations.

        .. TODO:: This could be improved.

        INPUT:

        - ``n`` -- positive integer

        - ``normalize`` -- boolean; remove gcd's during iteration

        OUTPUT: a projective dynamical system

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(CC,1)
            sage: f = DynamicalSystem_projective([x^2-y^2, x*y])
            sage: f.nth_iterate_map(3)
            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
                    (x^8 + (-7.00000000000000)*x^6*y^2 + 13.0000000000000*x^4*y^4 +
            (-7.00000000000000)*x^2*y^6 + y^8 : x^7*y + (-4.00000000000000)*x^5*y^3
            + 4.00000000000000*x^3*y^5 - x*y^7)

        ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: f = DynamicalSystem_projective([x^2-y^2, x*y, z^2+x^2])
            sage: f.nth_iterate_map(2)
            Dynamical System of Projective Space of dimension 2 over Integer Ring
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x^4 - 3*x^2*y^2 + y^4 : x^3*y - x*y^3 : 2*x^4 - 2*x^2*y^2 + y^4
            + 2*x^2*z^2 + z^4)

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: X = P.subscheme(x*z-y^2)
            sage: f = DynamicalSystem_projective([x^2, x*z, z^2], domain=X)
            sage: f.nth_iterate_map(2)
            Dynamical System of Closed subscheme of Projective Space of dimension
            2 over Rational Field defined by:
              -y^2 + x*z
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x^4 : x^2*z^2 : z^4)

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: f = DynamicalSystem_projective([y^2 * z^3, y^3 * z^2, x^5])
            sage: f.nth_iterate_map( 5, normalize=True)
            Dynamical System of Projective Space of dimension 2 over Rational
            Field
            Defn: Defined on coordinates by sending (x : y : z) to
            (y^202*z^443 : x^140*y^163*z^342 : x^645)
        """
        D = int(n)
        if D < 0:
            raise TypeError("iterate number must be a positive integer")
        if D == 1:
            return self
        H = End(self.domain())
        N = self.codomain().ambient_space().dimension_relative() + 1
        F = copy(self)
        Coord_ring = self.codomain().coordinate_ring()
        if isinstance(Coord_ring, QuotientRing_generic):
            PHI = H([Coord_ring.gen(i).lift() for i in range(N)])#makes a mapping
        else:
            PHI = H([Coord_ring.gen(i) for i in range(N)])
        while D:
            if D&1:
                PHI = PHI*F
                if normalize:
                    PHI.normalize_coordinates()
            if D > 1: #avoid extra iterate
                F = F*F
            if normalize:
                F.normalize_coordinates()
            D >>= 1
        return PHI.as_dynamical_system()

    def nth_iterate(self, P, n, **kwds):
        r"""
        Return the ``n``-th iterate of the point ``P`` by this
        dynamical system.

        If ``normalize`` is ``True``, then the coordinates are
        automatically normalized.

        .. TODO:: Is there a more efficient way to do this?

        INPUT:

        - ``P`` -- a point in this map's domain

        - ``n`` -- a positive integer

        kwds:

        - ``normalize`` -- (default: ``False``) boolean

        OUTPUT: a point in this map's codomain

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, 2*y^2])
            sage: Q = P(1,1)
            sage: f.nth_iterate(Q, 4, normalize=True)
            (1 : 1)

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x^2, 2*y^2, z^2-x^2])
            sage: Q = P(2,7,1)
            sage: f.nth_iterate(Q,2)
            (-16/7 : -2744 : 1)

        ::

            sage: R.<t> = PolynomialRing(QQ)
            sage: P.<x,y,z> = ProjectiveSpace(R,2)
            sage: f = DynamicalSystem_projective([x^2+t*y^2, (2-t)*y^2, z^2])
            sage: Q = P(2+t,7,t)
            sage: f.nth_iterate(Q,2)
            (t^4 + 2507*t^3 - 6787*t^2 + 10028*t + 16 : -2401*t^3 + 14406*t^2 -
            28812*t + 19208 : t^4)

        ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: X = P.subscheme(x^2-y^2)
            sage: f = DynamicalSystem_projective([x^2, y^2, z^2], domain=X)
            sage: f.nth_iterate(X(2,2,3), 3)
            (256 : 256 : 6561)

        ::

            sage: K.<c> = FunctionField(QQ)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^3 - 2*x*y^2 - c*y^3, x*y^2])
            sage: f.nth_iterate(P(c,1), 2)
            ((c^6 - 9*c^4 + 25*c^2 - c - 21)/(c^2 - 3) : 1)

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: H = Hom(P,P)
            sage: f = H([x^2+3*y^2, 2*y^2,z^2])
            sage: P(2, 7, 1).nth_iterate(f, -2)
            Traceback (most recent call last):
            ...
            TypeError: must be a forward orbit

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^3, x*y^2], domain=P)
            sage: f.nth_iterate(P(0, 1), 3, check=False)
            (0 : 0)
            sage: f.nth_iterate(P(0, 1), 3)
            Traceback (most recent call last):
            ...
            ValueError: [0, 0] does not define a valid point since all entries are 0

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
            sage: f = DynamicalSystem_projective([x^3, x*y^2], domain=P)
            sage: f.nth_iterate(P(2,1), 3, normalize=False)
            (134217728 : 524288)
            sage: f.nth_iterate(P(2,1), 3, normalize=True)
            (256 : 1)
        """
        n = ZZ(n)
        if n < 0:
            raise TypeError("must be a forward orbit")
        return self.orbit(P, [n,n+1], **kwds)[0]

    def degree_sequence(self, iterates=2):
        r"""
        Return sequence of degrees of normalized iterates starting with
        the degree of this dynamcial system.

        INPUT: ``iterates`` -- (default: 2) positive integer

        OUTPUT: list of integers

        EXAMPLES::

            sage: P2.<X,Y,Z> = ProjectiveSpace(QQ, 2)
            sage: f = DynamicalSystem_projective([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]

        ::

            sage: F.<t> = PolynomialRing(QQ)
            sage: P2.<X,Y,Z> = ProjectiveSpace(F, 2)
            sage: f = DynamicalSystem_projective([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: f = DynamicalSystem_projective([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: f = DynamicalSystem_projective([X*Y, Y*Z+Z^2, Z^2])
            sage: f.degree_sequence(10)
            [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
        """
        if int(iterates) < 1:
            raise TypeError("number of iterates must be a positive integer")

        if self.is_morphism():
            d = self.degree()
            D = [d**t for t in range(1, iterates+1)]
        else:
            F = self
            F.normalize_coordinates()
            D = [F.degree()]
            for n in range(2, iterates+1):
                F = F*self
                F.normalize_coordinates()
                D.append(F.degree())
        return D

    def dynamical_degree(self, N=3, prec=53):
        r"""
        Return an approximation to the dynamical degree of this dynamical
        system. The dynamical degree is defined as
        `\lim_{n \to \infty} \sqrt[n]{\deg(f^n)}`.

        INPUT:

        - ``N`` -- (default: 3) positive integer, iterate to use
          for approximation

        - ``prec`` -- (default: 53) positive integer, real precision
          to use when computing root

        OUTPUT: real number

        EXAMPLES::

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

        ::

            sage: P2.<X,Y,Z> = ProjectiveSpace(ZZ, 2)
            sage: f = DynamicalSystem_projective([X*Y, Y*Z+Z^2, Z^2])
            sage: f.dynamical_degree(N=5, prec=100)
            1.4309690811052555010452244131
        """
        if int(N) < 1:
            raise TypeError("number of iterates must be a positive integer")

        R = RealField(prec=prec)
        if self.is_morphism():
            return R(self.degree())
        else:
            D = self.nth_iterate_map(N, normalize=True).degree()
            return R(D).nth_root(N)

    def orbit(self, P, N, **kwds):
        r"""
        Return the orbit of the point ``P`` by this dynamical system.

        Let `F` be this dynamical system. If ``N`` is an integer return
        `[P,F(P),\ldots,F^N(P)]`. If ``N`` is a list or tuple `N=[m,k]`
        return `[F^m(P),\ldots,F^k(P)]`.
        Automatically normalize the points if ``normalize=True``. Perform
        the checks on point initialization if ``check=True``.

        INPUT:

        - ``P`` -- a point in this dynamical system's domain

        - ``n`` -- a non-negative integer or list or tuple of two
          non-negative integers

        kwds:

        - ``check`` --  (default: ``True``) boolean

        - ``normalize`` -- (default: ``False``) boolean

        OUTPUT: a list of points in this dynamical system's codomain

        EXAMPLES::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2-z^2, 2*z^2])
            sage: f.orbit(P(1,2,1), [2,4])
            [(34 : 5 : 8), (1181 : -39 : 128), (1396282 : -14863 : 32768)]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: X = P.subscheme(x^2-y^2)
            sage: f = DynamicalSystem_projective([x^2, y^2, x*z], domain=X)
            sage: f.orbit(X(2,2,3), 3, normalize=True)
            [(2 : 2 : 3), (2 : 2 : 3), (2 : 2 : 3), (2 : 2 : 3)]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2])
            sage: f.orbit(P.point([1,2],False), 4, check=False)
            [(1 : 2), (5 : 4), (41 : 16), (1937 : 256), (3817505 : 65536)]

        ::

            sage: K.<c> = FunctionField(QQ)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2+c*y^2, y^2])
            sage: f.orbit(P(0,1), 3)
            [(0 : 1), (c : 1), (c^2 + c : 1), (c^4 + 2*c^3 + c^2 + c : 1)]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2,y^2], domain=P)
            sage: f.orbit(P.point([1, 2], False), 4, check=False)
            [(1 : 2), (5 : 4), (41 : 16), (1937 : 256), (3817505 : 65536)]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2, 2*y^2], domain=P)
            sage: P(2, 1).orbit(f,[-1, 4])
            Traceback (most recent call last):
            ...
            TypeError: orbit bounds must be non-negative
            sage: P(2, 1).orbit(f, 0.1)
            Traceback (most recent call last):
            ...
            TypeError: Attempt to coerce non-integral RealNumber to Integer

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^3, x*y^2], domain=P)
            sage: P(0, 1).orbit(f, 3)
            Traceback (most recent call last):
            ...
            ValueError: [0, 0] does not define a valid point since all entries are 0
            sage: P(0, 1).orbit(f, 3, check=False)
            [(0 : 1), (0 : 0), (0 : 0), (0 : 0)]

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
            sage: f = DynamicalSystem_projective([x^3, x*y^2], domain=P)
            sage: P(2,1).orbit(f, 3, normalize=False)
            [(2 : 1), (8 : 2), (512 : 32), (134217728 : 524288)]
            sage: P(2, 1).orbit(f, 3, normalize=True)
            [(2 : 1), (4 : 1), (16 : 1), (256 : 1)]
        """
        if not isinstance(N,(list,tuple)):
            N = [0,N]
        N[0] = ZZ(N[0])
        N[1] = ZZ(N[1])
        if N[0] < 0 or N[1] < 0:
            raise TypeError("orbit bounds must be non-negative")
        if N[0] > N[1]:
            return([])

        Q = P
        check = kwds.pop("check",True)
        normalize = kwds.pop("normalize",False)

        if normalize:
            Q.normalize_coordinates()
        for i in range(1, N[0]+1):
            Q = self(Q, check)
            if normalize:
                Q.normalize_coordinates()
        orb = [Q]
        for i in range(N[0]+1, N[1]+1):
            Q = self(Q, check)
            if normalize:
                Q.normalize_coordinates()
            orb.append(Q)
        return(orb)

    def resultant(self, normalize=False):
        r"""
        Computes the resultant of the defining polynomials of
        this dynamical system.

        If ``normalize`` is ``True``, then first normalize the coordinate
        functions with :meth:`normalize_coordinates`.

        INPUT:

        - ``normalize`` -- (default: ``False``) boolean

        OUTPUT: an element of the base ring of this map

        EXAMPLES::

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

        ::

            sage: R.<t> = PolynomialRing(GF(17))
            sage: P.<x,y> = ProjectiveSpace(R,1)
            sage: f = DynamicalSystem_projective([t*x^2+t*y^2, 6*y^2])
            sage: f.resultant()
            2*t^2

        ::

            sage: R.<t> = PolynomialRing(GF(17))
            sage: P.<x,y,z> = ProjectiveSpace(R,2)
            sage: f = DynamicalSystem_projective([t*x^2+t*y^2, 6*y^2, 2*t*z^2])
            sage: f.resultant()
            13*t^8

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: F = DynamicalSystem_projective([x^2+y^2,6*y^2,10*x*z+z^2+y^2])
            sage: F.resultant()
            1296

        ::

            sage: R.<t>=PolynomialRing(QQ)
            sage: s = (t^3+t+1).roots(QQbar)[0][0]
            sage: P.<x,y>=ProjectiveSpace(QQbar,1)
            sage: f = DynamicalSystem_projective([s*x^3-13*y^3, y^3-15*y^3])
            sage: f.resultant()
            871.6925062959149?
            """
        if normalize:
            F = copy(self)
            F.normalize_coordinates()
        else:
            F = self

        if self.domain().dimension_relative() == 1:
            x = self.domain().gen(0)
            y = self.domain().gen(1)
            d = self.degree()
            f = F[0].substitute({y:1})
            g = F[1].substitute({y:1})
            #Try to use pari first, as it is faster for one dimensional case
            #however the coercion from a Pari object to a sage object breaks
            #in the case of QQbar, so we just pass it into the macaulay resultant
            try:
                res = (f.lc() ** (d - g.degree()) * g.lc() ** (d - f.degree())
                       * f.__pari__().polresultant(g, x))
                return(self.domain().base_ring()(res))
            except (TypeError, PariError):
                pass
        #Otherwise, use Macaulay
        R = F[0].parent()
        res = R.macaulay_resultant(list(F._polys))
        return res #Coercion here is not necessary as it is already done in Macaulay Resultant

    @cached_method
    def primes_of_bad_reduction(self, check=True):
        r"""
        Determine the primes of bad reduction for this dynamical system.

        Must be defined over a number field.

        If ``check`` is ``True``, each prime is verified to be of
        bad reduction.

        ALGORITHM:

        `p` is a prime of bad reduction if and only if the defining
        polynomials of self have a common zero. Or stated another way,
        `p` is a prime of bad reduction if and only if the radical of
        the ideal defined by the defining polynomials of self is not
        `(x_0,x_1,\ldots,x_N)`.  This happens if and only if some
        power of each `x_i` is not in the ideal defined by the
        defining polynomials of self. This last condition is what is
        checked. The lcm of the coefficients of the monomials `x_i` in
        a Groebner basis is computed. This may return extra primes.

        INPUT:

        - ``check`` -- (default: ``True``) boolean

        OUTPUT: a list of primes

        EXAMPLES::

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

        ::

            sage: P.<x,y,z,w> = ProjectiveSpace(QQ,3)
            sage: f = DynamicalSystem_projective([12*x*z-7*y^2, 31*x^2-y^2, 26*z^2, 3*w^2-z*w])
            sage: f.primes_of_bad_reduction()
            [2, 3, 7, 13, 31]

        A number field example::

            sage: R.<z> = QQ[]
            sage: K.<a> = NumberField(z^2 - 2)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([1/3*x^2+1/a*y^2, y^2])
            sage: f.primes_of_bad_reduction()
            [Fractional ideal (a), Fractional ideal (3)]

        This is an example where check = False returns extra primes::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: f = DynamicalSystem_projective([3*x*y^2 + 7*y^3 - 4*y^2*z + 5*z^3,
            ....:                                 -5*x^3 + x^2*y + y^3 + 2*x^2*z,
            ....:                                 -2*x^2*y + x*y^2 + y^3 - 4*y^2*z + x*z^2])
            sage: f.primes_of_bad_reduction(False)
            [2, 5, 37, 2239, 304432717]
            sage: f.primes_of_bad_reduction()
            [5, 37, 2239, 304432717]
        """
        if (not is_ProjectiveSpace(self.domain())) or (not is_ProjectiveSpace(self.codomain())):
            raise NotImplementedError("not implemented for subschemes")
        K = FractionField(self.codomain().base_ring())
        #The primes of bad reduction are the support of the resultant for number fields

        if K in NumberFields():
            if K != QQ:
                F = copy(self)
                F.normalize_coordinates()
                return (K(F.resultant()).support())
            else:
                #For the rationals, we can use groebner basis, as it is quicker in practice
                R = self.coordinate_ring()
                F = self._polys

                if R.base_ring().is_field():
                    J = R.ideal(F)
                else:
                    S = PolynomialRing(R.base_ring().fraction_field(), R.gens(), R.ngens())
                    J = S.ideal([S.coerce(F[i]) for i in range(R.ngens())])
                if J.dimension() > 0:
                    raise TypeError("not a morphism")
                #normalize to coefficients in the ring not the fraction field.
                F = [F[i] * lcm([F[j].denominator() for j in range(len(F))]) for i in range(len(F))]

                #move the ideal to the ring of integers
                if R.base_ring().is_field():
                    S = PolynomialRing(R.base_ring().ring_of_integers(), R.gens(), R.ngens())
                    F = [F[i].change_ring(R.base_ring().ring_of_integers()) for i in range(len(F))]
                    J = S.ideal(F)
                else:
                    J = R.ideal(F)
                GB = J.groebner_basis()
                badprimes = []

                #get the primes dividing the coefficients of the monomials x_i^k_i
                for i in range(len(GB)):
                    LT = GB[i].lt().degrees()
                    power = 0
                    for j in range(R.ngens()):
                        if LT[j] != 0:
                            power += 1
                    if power == 1:
                        badprimes = badprimes + GB[i].lt().coefficients()[0].support()
                badprimes = sorted(set(badprimes))

                #check to return only the truly bad primes
                if check:
                    index = 0
                    while index < len(badprimes):  #figure out which primes are really bad primes...
                        S = PolynomialRing(GF(badprimes[index]), R.gens(), R.ngens())
                        J = S.ideal([S.coerce(F[j]) for j in range(R.ngens())])
                        if J.dimension() == 0:
                            badprimes.pop(index)
                        else:
                            index += 1
                return(badprimes)
        else:
            raise TypeError("base ring must be number field or number field ring")

    def conjugate(self, M):
        r"""
        Conjugate this dynamical system by ``M``, i.e. `M^{-1} \circ f \circ M`.

        If possible the new map will be defined over the same space.
        Otherwise, will try to coerce to the base ring of ``M``.

        INPUT:

        - ``M`` -- a square invertible matrix

        OUTPUT: a dynamical system

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2])
            sage: f.conjugate(matrix([[1,2], [0,1]]))
            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)

        ::

            sage: R.<x> = PolynomialRing(QQ)
            sage: K.<i> = NumberField(x^2+1)
            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^3+y^3, y^3])
            sage: f.conjugate(matrix([[i,0], [0,-i]]))
            Dynamical System of Projective Space of dimension 1 over Integer Ring
              Defn: Defined on coordinates by sending (x : y) to
                    (-x^3 + y^3 : -y^3)

        ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: f = DynamicalSystem_projective([x^2+y^2 ,y^2, y*z])
            sage: f.conjugate(matrix([[1,2,3], [0,1,2], [0,0,1]]))
            Dynamical System of Projective Space of dimension 2 over Integer Ring
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x^2 + 4*x*y + 3*y^2 + 6*x*z + 9*y*z + 7*z^2 : y^2 + 2*y*z : y*z + 2*z^2)

        ::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2])
            sage: f.conjugate(matrix([[2,0], [0,1/2]]))
            Dynamical System of Projective Space of dimension 1 over Rational Field
              Defn: Defined on coordinates by sending (x : y) to
                    (2*x^2 + 1/8*y^2 : 1/2*y^2)

        ::

            sage: R.<x> = PolynomialRing(QQ)
            sage: K.<i> = NumberField(x^2+1)
            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([1/3*x^2+1/2*y^2, y^2])
            sage: f.conjugate(matrix([[i,0], [0,-i]]))
            Dynamical System of Projective Space of dimension 1 over Number Field in i with defining polynomial x^2 + 1
              Defn: Defined on coordinates by sending (x : y) to
                    ((1/3*i)*x^2 + (1/2*i)*y^2 : (-i)*y^2)
        """
        if not (M.is_square() == 1 and M.determinant() != 0
            and M.ncols() == self.domain().ambient_space().dimension_relative() + 1):
            raise TypeError("matrix must be invertible and size dimension + 1")
        X = M * vector(self[0].parent().gens())
        F = vector(self._polys)
        F = F(list(X))
        N = M.inverse()
        F = N * F
        R = self.codomain().ambient_space().coordinate_ring()
        try:
            F = [R(f) for f in F]
            PS = self.codomain()
        except TypeError: #no longer defined over same ring
            R = R.change_ring(M.base_ring())
            F = [R(f) for f in F]
            PS = self.codomain().change_ring(M.base_ring())
        return DynamicalSystem_projective(F, domain=PS)

    def green_function(self, P, v, **kwds):
        r"""
        Evaluate the local Green's function at the place ``v`` for ``P``
        with ``N`` terms of the series or to within a given error bound.

        Must be over a number field or order of a number field. Note that
        this is the absolute local Green's function so is scaled by the
        degree of the base field.

        Use ``v=0`` for the archimedean place over `\QQ` or field embedding.
        Non-archimedean places are prime ideals for number fields or primes
        over `\QQ`.

        ALGORITHM:

        See Exercise 5.29 and Figure 5.6 of [Sil2007]_.

        INPUT:

        - ``P`` -- a projective point

        - ``v`` -- non-negative integer. a place, use ``0`` for the
          archimedean place

        kwds:

        - ``N`` -- (optional - default: 10) positive integer. number of
          terms of the series to use

        - ``prec`` -- (default: 100) positive integer, float point or
          `p`-adic precision

        - ``error_bound`` -- (optional) a positive real number

        OUTPUT: a real number

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, x*y]);
            sage: Q = P(5, 1)
            sage: f.green_function(Q, 0, N=200, prec=200)
            1.6460930160038721802875250367738355497198064992657997569827

        ::

            sage: K.<w> = QuadraticField(3)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([17*x^2+1/7*y^2, 17*w*x*y])
            sage: f.green_function(P.point([w, 2], False), K.places()[1])
            1.7236334013785676107373093775
            sage: f.green_function(P([2, 1]), K.ideal(7), N=7)
            0.48647753726382832627633818586
            sage: f.green_function(P([w, 1]), K.ideal(17), error_bound=0.001)
            -0.70813041039490996737374178059

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, x*y])
            sage: f.green_function(P.point([5,2], False), 0, N=30)
            1.7315451844777407992085512000
            sage: f.green_function(P.point([2,1], False), 0, N=30)
            0.86577259223181088325226209926
            sage: f.green_function(P.point([1,1], False), 0, N=30)
            0.43288629610862338612700146098
        """
        N = kwds.get('N', 10)                       #Get number of iterates (if entered)
        err = kwds.get('error_bound', None)         #Get error bound (if entered)
        prec = kwds.get('prec', 100)                #Get precision (if entered)
        R = RealField(prec)
        localht = R(0)
        BR = FractionField(P.codomain().base_ring())
        GBR = self.change_ring(BR) #so the heights work

        if not BR in NumberFields():
            raise NotImplementedError("must be over a number field or a number field order")
        if not BR.is_absolute():
            raise TypeError("must be an absolute field")

        #For QQ the 'flip-trick' works better over RR or Qp
        if isinstance(v, (NumberFieldFractionalIdeal, RingHomomorphism_im_gens)):
            K = BR
        elif is_prime(v):
            K = Qp(v, prec)
        elif v == 0:
            K = R
            v = BR.places(prec=prec)[0]
        else:
            raise ValueError("invalid valuation (=%s) entered"%v)

        #Coerce all polynomials in F into polynomials with coefficients in K
        F = self.change_ring(K, check=False)
        d = F.degree()
        dim = F.codomain().ambient_space().dimension_relative()
        Q = P.change_ring(K, check=False)

        if err is not None:
            err = R(err)
            if not err > 0:
                raise ValueError("error bound (=%s) must be positive"%err)

            #if doing error estimates, compute needed number of iterates
            D = (dim + 1) * (d - 1) + 1
            #compute upper bound
            if isinstance(v, RingHomomorphism_im_gens): #archimedean
                vindex = BR.places(prec=prec).index(v)
                U = GBR.local_height_arch(vindex, prec=prec) + R(binomial(dim + d, d)).log()
            else: #non-archimedean
                U = GBR.local_height(v, prec=prec)

            #compute lower bound - from explicit polynomials of Nullstellensatz
            CR = GBR.codomain().ambient_space().coordinate_ring() #.lift() only works over fields
            I = CR.ideal(GBR.defining_polynomials())
            maxh = 0
            Res = 1
            for k in range(dim + 1):
                CoeffPolys = (CR.gen(k) ** D).lift(I)
                h = 1
                for poly in CoeffPolys:
                    if poly != 0:
                        for c in poly.coefficients():
                            Res = lcm(Res, c.denominator())
                for poly in CoeffPolys:
                    if poly != 0:
                        if isinstance(v, RingHomomorphism_im_gens): #archimedean
                            if BR == QQ:
                                h = max([(Res*c).local_height_arch(prec=prec) for c in poly.coefficients()])
                            else:
                                h = max([(Res*c).local_height_arch(vindex, prec=prec) for c in poly.coefficients()])
                        else: #non-archimedean
                            h = max([c.local_height(v, prec=prec) for c in poly.coefficients()])
                        if h > maxh:
                            maxh=h
            if maxh == 0:
                maxh = 1  #avoid division by 0
            if isinstance(v, RingHomomorphism_im_gens): #archimedean
                L = R(Res / ((dim + 1) * binomial(dim + D - d, D - d) * maxh)).log().abs()
            else: #non-archimedean
                L = R(Res / maxh).log().abs()
            C = max([U, L])
            if C != 0:
                N = R(C / (err*(d-1))).log(d).abs().ceil()
            else: #we just need log||P||_v
                N=1

        #START GREEN FUNCTION CALCULATION
        if isinstance(v, RingHomomorphism_im_gens):  #embedding for archimedean local height
            for i in range(N+1):
                Qv = [ (v(t).abs()) for t in Q ]
                m = -1
                #compute the maximum absolute value of entries of a, and where it occurs
                for n in range(dim + 1):
                    if Qv[n] > m:
                        j = n
                        m = Qv[n]
                # add to sum for the Green's function
                localht += ((1/R(d))**R(i)) * (R(m).log())
                #get the next iterate
                if i < N:
                    Q.scale_by(1/Q[j])
                    Q = F(Q, False)
            return (1/BR.absolute_degree()) * localht

        #else - prime or prime ideal for non-archimedean
        for i in range(N + 1):
            if BR == QQ:
                Qv = [ R(K(t).abs()) for t in Q ]
            else:
                Qv = [ R(t.abs_non_arch(v)) for t in Q ]
            m = -1
            #compute the maximum absolute value of entries of a, and where it occurs
            for n in range(dim + 1):
                if Qv[n] > m:
                    j = n
                    m = Qv[n]
            # add to sum for the Green's function
            localht += (1/R(d))**R(i) * (R(m).log())
            #get the next iterate
            if i < N:
                Q.scale_by(1 / Q[j])
                Q = F(Q, False)
        return (1 / BR.absolute_degree()) * localht

    def canonical_height(self, P, **kwds):
        r"""
        Evaluate the (absolute) canonical height of ``P`` with respect to
        this dynamical system.

        Must be over number field or order of a number field. Specify
        either the number of terms of the series to evaluate or the
        error bound required.

        ALGORITHM:

        The sum of the Green's function at the archimedean places and
        the places of bad reduction.

        If function is defined over `\QQ` uses Wells' Algorithm, which
        allows us to not have to factor the resultant.

        INPUT:

        - ``P`` -- a projective point

        kwds:

        - ``badprimes`` -- (optional) a list of primes of bad reduction

        - ``N`` -- (default: 10) positive integer. number of
          terms of the series to use in the local green functions

        - ``prec`` -- (default: 100) positive integer, float point or
          `p`-adic precision

        - ``error_bound`` -- (optional) a positive real number

        OUTPUT: a real number

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, 2*x*y]);
            sage: f.canonical_height(P.point([5,4]), error_bound=0.001)
            2.1970553519503404898926835324
            sage: f.canonical_height(P.point([2,1]), error_bound=0.001)
            1.0984430632822307984974382955

        Notice that preperiodic points may not return exactly 0::

            sage: R.<X> = PolynomialRing(QQ)
            sage: K.<a> = NumberField(X^2 + X - 1)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2-2*y^2, y^2])
            sage: Q = P.point([a,1])
            sage: f.canonical_height(Q, error_bound=0.000001) # Answer only within error_bound of 0
            5.7364919788790160119266380480e-8
            sage: f.nth_iterate(Q,2) == Q # but it is indeed preperiodic
            True

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: X = P.subscheme(x^2-y^2);
            sage: f = DynamicalSystem_projective([x^2,y^2, 4*z^2], domain=X);
            sage: Q = X([4,4,1])
            sage: f.canonical_height(Q, badprimes=[2])
            0.0013538030870311431824555314882

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: X = P.subscheme(x^2-y^2);
            sage: f = DynamicalSystem_projective([x^2,y^2, 30*z^2], domain=X)
            sage: Q = X([4, 4, 1])
            sage: f.canonical_height(Q, badprimes=[2,3,5], prec=200)
            2.7054056208276961889784303469356774912979228770208655455481

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([1000*x^2-29*y^2, 1000*y^2])
            sage: Q = P(-1/4, 1)
            sage: f.canonical_height(Q, error_bound=0.01)
            3.7996079979254623065837411853

        ::

            sage: RSA768 = 123018668453011775513049495838496272077285356959533479219732245215\
            ....: 1726400507263657518745202199786469389956474942774063845925192557326303453731548\
            ....: 2685079170261221429134616704292143116022212404792747377940806653514195974598569\
            ....: 02143413
            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([RSA768*x^2 + y^2, x*y])
            sage: Q = P(RSA768,1)
            sage: f.canonical_height(Q, error_bound=0.00000000000000001)
            931.18256422718241278672729195
        """
        bad_primes = kwds.get("badprimes", None)
        prec = kwds.get("prec", 100)
        error_bound = kwds.get("error_bound", None)
        K = FractionField(self.codomain().base_ring())

        if not K in NumberFields():
            if not K is QQbar:
                raise NotImplementedError("must be over a number field or a number field order or QQbar")
            else:
                #since this an absolute height, we can compute the height of a QQbar point
                #by choosing any number field it is defined over.
                Q = P._number_field_from_algebraics()
                K = Q.codomain().base_ring()
                f = self._number_field_from_algebraics().as_dynamical_system()
                if K == QQ:
                    K = f.base_ring()
                    Q = Q.change_ring(K)
                elif f.base_ring() == QQ:
                    f = f.change_ring(K)
                else:
                    K, phi, psi, b = K.composite_fields(f.base_ring(), both_maps=True)[0]
                    Q = Q.change_ring(K, embedding=phi)
                    f = f.change_ring(K, embedding=psi)
        else:
            if not K.is_absolute():
                raise TypeError("must be an absolute field")
            Q = P
            f = self

        # After moving from QQbar to K being something like QQ, we need
        # to renormalize f, especially to match the normalized resultant.
        f.normalize_coordinates()

        # If our map and point are defined on P^1(QQ), use Wells' Algorithm
        # instead of the usual algorithm using local Green's functions:
        if K is QQ and self.codomain().ambient_space().dimension_relative() == 1:
            # write our point with coordinates whose gcd is 1
            Q.normalize_coordinates()
            if Q.parent().value_ring() is QQ:
                Q.clear_denominators()
            #assures integer coeffcients
            coeffs = f[0].coefficients() + f[1].coefficients()
            t = 1
            for c in coeffs:
                t = lcm(t, c.denominator())
            A = t * f[0]
            B = t * f[1]
            Res = f.resultant(normalize=True)
            H = 0
            x_i = Q[0]
            y_i = Q[1]
            d = self.degree()
            R = RealField(prec)
            N = kwds.get('N', 10)
            err = kwds.get('error_bound', None)
            #computes the error bound as defined in Algorithm 3.1 of [WELLS]
            if Res > 1:
                if not err is None:
                    err = err / 2
                    N = ceil((R(Res.abs()).log().log() - R(d-1).log() - R(err).log())/(R(d).log()))
                    if N < 1:
                        N = 1
                    kwds.update({'error_bound': err})
                    kwds.update({'N': N})
                for n in range(N):
                    x = A(x_i,y_i) % Res**(N-n)
                    y = B(x_i,y_i) % Res**(N-n)
                    g = gcd([x, y, Res])
                    H = H + R(g).abs().log() / (d**(n+1))
                    x_i = x / g
                    y_i = y / g
            # this looks different than Wells' Algorithm because of the difference
            # between what Wells' calls H_infty,
            # and what Green's Function returns for the infinite place
            h = f.green_function(Q, 0 , **kwds) - H + R(t).log()
            # The value returned by Well's algorithm may be negative. As the canonical height
            # is always nonnegative, so if this value is within -err of 0, return 0.
            if h < 0:
                assert h > -err, "A negative height less than -error_bound was computed. " + \
                 "This should be impossible, please report bug on trac.sagemath.org."
                    # This should be impossible. The error bound for Wells' is rigorous
                    # and the actual height is always >= 0. If we see something less than -err,
                    # something has g one very wrong.
                h = R(0)
            return h

        if bad_primes is None:
            bad_primes = []
            for b in Q:
                if K == QQ:
                    bad_primes += b.denominator().prime_factors()
                else:
                    bad_primes += b.denominator_ideal().prime_factors()
            bad_primes += K(f.resultant(normalize=True)).support()
            bad_primes = list(set(bad_primes))

        emb = K.places(prec=prec)
        num_places = len(emb) + len(bad_primes)
        if not error_bound is None:
            error_bound /= num_places
        R = RealField(prec)
        h = R.zero()

        ##update the keyword dictionary for use in green_function
        kwds.update({"badprimes": bad_primes})
        kwds.update({"error_bound": error_bound})

        # Archimedean local heights
        # :: WARNING: If places is fed the default Sage precision of 53 bits,
        # it uses Real or Complex Double Field in place of RealField(prec) or ComplexField(prec)
        # the function is_RealField does not identify RDF as real, so we test for that ourselves.
        for v in emb:
            if is_RealField(v.codomain()) or v.codomain() is RDF:
                dv = R.one()
            else:
                dv = R(2)
            h += dv * f.green_function(Q, v, **kwds)     #arch Green function

        # Non-Archimedean local heights
        for v in bad_primes:
            if K == QQ:
                dv = R.one()
            else:
                dv = R(v.residue_class_degree() * v.absolute_ramification_index())
            h += dv * f.green_function(Q, v, **kwds)  #non-arch Green functions
        return h

    def height_difference_bound(self, prec=None):
        r"""
        Return an upper bound on the different between the canonical
        height of a point with respect to this dynamical system and the
        absolute height of the point.

        This map must be a morphism.

        ALGORITHM:

        Uses a Nullstellensatz argument to compute the constant.
        For details: see [Hutz2015]_.

        INPUT:

        - ``prec`` -- (default: :class:`RealField` default)
          positive integer, float point precision

        OUTPUT: a real number

        EXAMPLES::

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

        This function does not automatically normalize. ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: f = DynamicalSystem_projective([4*x^2+100*y^2, 210*x*y, 10000*z^2])
            sage: f.height_difference_bound()
            11.0020998412042
            sage: f.normalize_coordinates()
            sage: f.height_difference_bound()
            10.3089526606443

       A number field example::

            sage: R.<x> = QQ[]
            sage: K.<c> = NumberField(x^3 - 2)
            sage: P.<x,y,z> = ProjectiveSpace(K,2)
            sage: f = DynamicalSystem_projective([1/(c+1)*x^2+c*y^2, 210*x*y, 10000*z^2])
            sage: f.height_difference_bound()
            11.0020998412042

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQbar,2)
            sage: f = DynamicalSystem_projective([x^2, QQbar(sqrt(-1))*y^2, QQbar(sqrt(3))*z^2])
            sage: f.height_difference_bound()
            3.43967790223022
        """
        FF = FractionField(self.domain().base_ring()) #lift will only work over fields, so coercing into FF
        if not FF in NumberFields():
            if FF == QQbar:
                #since this is absolute height, we can choose any number field over which the
                #function is defined.
                f = self._number_field_from_algebraics()
            else:
                raise NotImplementedError("fraction field of the base ring must be a number field or QQbar")
        else:
            f = self.change_ring(FF)
        if prec is None:
            R = RealField()
        else:
            R = RealField(prec)
        N = f.domain().dimension_relative()
        d = f.degree()
        D = (N + 1) * (d - 1) + 1
        #compute upper bound
        U = f.global_height(prec) + R(binomial(N + d, d)).log()
        #compute lower bound - from explicit polynomials of Nullstellensatz
        CR = f.domain().coordinate_ring()
        I = CR.ideal(f.defining_polynomials())
        MCP = []
        for k in range(N + 1):
            CoeffPolys = (CR.gen(k) ** D).lift(I)
            Res = lcm([1] + [abs(coeff.denominator()) for val in CoeffPolys
                             for coeff in val.coefficients()])
            h = max([c.global_height() for g in CoeffPolys for c in (Res*g).coefficients()])
            MCP.append([Res, h]) #since we need to clear denominators
        maxh = 0
        gcdRes = 0
        for val in MCP:
            gcdRes = gcd(gcdRes, val[0])
            maxh = max(maxh, val[1])
        L = abs(R(gcdRes).log() - R((N + 1) * binomial(N + D - d, D - d)).log() - maxh)
        C = max(U, L) #height difference dh(P) - L <= h(f(P)) <= dh(P) +U
        return(C / (d - 1))

    def multiplier(self, P, n, check=True):
        r"""
        Return the multiplier of the point ``P`` of period ``n`` with
        respect to this dynamical system.

        INPUT:

        - ``P`` -- a point on domain of this map

        - ``n`` -- a positive integer, the period of ``P``

        - ``check`` -- (default: ``True``) boolean; verify that ``P``
          has period ``n``

        OUTPUT:

        A square matrix of size ``self.codomain().dimension_relative()``
        in the ``base_ring`` of this dynamical system.

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([7*x^2 - 28*y^2, 24*x*y])
            sage: f.multiplier(P(2,5), 4)
            [231361/20736]

        ::

            sage: P.<x,y> = ProjectiveSpace(CC,1)
            sage: f = DynamicalSystem_projective([x^3 - 25*x*y^2 + 12*y^3, 12*y^3])
            sage: f.multiplier(P(1,1), 5)
            [0.389017489711935]

        ::

            sage: P.<x,y> = ProjectiveSpace(RR,1)
            sage: f = DynamicalSystem_projective([x^2-2*y^2, y^2])
            sage: f.multiplier(P(2,1), 1)
            [4.00000000000000]

        ::

            sage: P.<x,y> = ProjectiveSpace(Qp(13),1)
            sage: f = DynamicalSystem_projective([x^2-29/16*y^2, y^2])
            sage: f.multiplier(P(5,4), 3)
            [6 + 8*13 + 13^2 + 8*13^3 + 13^4 + 8*13^5 + 13^6 + 8*13^7 + 13^8 +
            8*13^9 + 13^10 + 8*13^11 + 13^12 + 8*13^13 + 13^14 + 8*13^15 + 13^16 +
            8*13^17 + 13^18 + 8*13^19 + O(13^20)]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2-y^2, y^2])
            sage: f.multiplier(P(0,1), 1)
            Traceback (most recent call last):
            ...
            ValueError: (0 : 1) is not periodic of period 1
        """
        if check:
            if self.nth_iterate(P, n) != P:
                raise ValueError("%s is not periodic of period %s"%(P, n))
            if n < 1:
                raise ValueError("period must be a positive integer")
        N = self.domain().ambient_space().dimension_relative()
        l = identity_matrix(FractionField(self.codomain().base_ring()), N, N)
        Q = P
        Q.normalize_coordinates()
        index = N
        indexlist = [] #keep track of which dehomogenizations are needed
        while Q[index] == 0:
            index -= 1
        indexlist.append(index)
        for i in range(0, n):
            F = []
            R = self(Q)
            R.normalize_coordinates()
            index = N
            while R[index] == 0:
                index -= 1
            indexlist.append(index)
            #dehomogenize and compute multiplier
            F = self.dehomogenize((indexlist[i],indexlist[i+1]))
            #get the correct order for chain rule matrix multiplication
            l = F.jacobian()(tuple(Q.dehomogenize(indexlist[i])))*l
            Q = R
        return l

    def _multipliermod(self, P, n, p, k):
        r"""
        Return the multiplier of the point ``P`` of period ``n`` with
        respect to this dynamical system modulo `p^k`.

        This map must be an endomorphism of projective space defined
        over `\QQ` or `\ZZ`. This function should not be used at the top
        level as it does not perform input checks. It is used primarily
        for the rational preperiodic and periodic point algorithms.

        INPUT:

        - ``P`` -- a point on domain of this map

        - ``n`` -- a positive integer, the period of ``P``

        - ``p`` -- a positive integer

        - ``k`` -- a positive integer

        OUTPUT:

        A square matrix of size ``self.codomain().dimension_relative()``
        in `\ZZ/(p^k)\ZZ`.

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2-29/16*y^2, y^2])
            sage: f._multipliermod(P(5,4), 3, 11, 1)
            [3]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2-29/16*y^2, y^2])
            sage: f._multipliermod(P(5,4), 3, 11, 2)
            [80]
        """
        N = self.domain().dimension_relative()
        BR = FractionField(self.codomain().base_ring())
        l = identity_matrix(BR, N, N)
        Q = copy(P)
        g = gcd(Q._coords) #we can't use normalize_coordinates since it can cause denominators
        Q.scale_by(1 / g)
        index = N
        indexlist = [] #keep track of which dehomogenizations are needed
        while Q[index] % p == 0:
            index -= 1
        indexlist.append(index)
        for i in range(0, n):
            F = []
            R = self(Q, False)
            g = gcd(R._coords)
            R.scale_by(1 / g)
            R_list = list(R)
            for index in range(N + 1):
                R_list[index] = R_list[index] % (p ** k)
            R._coords = tuple(R_list)
            index = N
            while R[index] % p == 0:
                index -= 1
            indexlist.append(index)
            #dehomogenize and compute multiplier
            F = self.dehomogenize((indexlist[i],indexlist[i+1]))
            l = (F.jacobian()(tuple(Q.dehomogenize(indexlist[i])))*l) % (p ** k)
            Q = R
        return(l)

    def possible_periods(self, **kwds):
        r"""
        Return the set of possible periods for rational periodic points of
        this dynamical system.

        Must be defined over `\ZZ` or `\QQ`.

        ALGORITHM:

        Calls ``self.possible_periods()`` modulo all primes of good reduction
        in range ``prime_bound``. Return the intersection of those lists.

        INPUT:

        kwds:

        - ``prime_bound`` --  (default: ``[1, 20]``) a list or tuple of
           two positive integers or an integer for the upper bound

        - ``bad_primes`` -- (optional) a list or tuple of integer primes,
          the primes of bad reduction

        - ``ncpus`` -- (default: all cpus) number of cpus to use in parallel

        OUTPUT: a list of positive integers

        EXAMPLES::

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

        ::

            sage: PS.<x,y> = ProjectiveSpace(1,QQ)
            sage: f = DynamicalSystem_projective([5*x^3 - 53*x*y^2 + 24*y^3, 24*y^3])
            sage: f.possible_periods(prime_bound=[1,5])
            Traceback (most recent call last):
            ...
            ValueError: no primes of good reduction in that range
            sage: f.possible_periods(prime_bound=[1,10])
            [1, 4, 12]
            sage: f.possible_periods(prime_bound=[1,20])
            [1, 4]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: f = DynamicalSystem_projective([2*x^3 - 50*x*z^2 + 24*z^3,
            ....:                                 5*y^3 - 53*y*z^2 + 24*z^3, 24*z^3])
            sage: f.possible_periods(prime_bound=10)
            [1, 2, 6, 20, 42, 60, 140, 420]
            sage: f.possible_periods(prime_bound=20) # long time
            [1, 20]
        """
        if self.domain().base_ring() not in [ZZ, QQ]:
            raise NotImplementedError("must be ZZ or QQ")

        primebound = kwds.pop("prime_bound", [1, 20])
        badprimes = kwds.pop("bad_primes", None)
        num_cpus = kwds.pop("ncpus", ncpus())

        if not isinstance(primebound, (list, tuple)):
            try:
                primebound = [1, ZZ(primebound)]
            except TypeError:
                raise TypeError("prime bound must be an integer")
        else:
            try:
                primebound[0] = ZZ(primebound[0])
                primebound[1] = ZZ(primebound[1])
            except TypeError:
                raise TypeError("prime bounds must be integers")

        if badprimes is None:
            badprimes = self.primes_of_bad_reduction()

        firstgood = 0

        def parallel_function(morphism):
            return morphism.possible_periods()

        # Calling possible_periods for each prime in parallel
        parallel_data = []
        for q in primes(primebound[0], primebound[1] + 1):
            if not (q in badprimes):
                F = self.change_ring(GF(q))
                parallel_data.append(((F,), {}))

        parallel_iter = p_iter_fork(num_cpus, 0)
        parallel_results = list(parallel_iter(parallel_function, parallel_data))

        for result in parallel_results:
            possible_periods = result[1]
            if firstgood == 0:
                periods = set(possible_periods)
                firstgood = 1
            else:
                periodsq = set(possible_periods)
                periods = periods.intersection(periodsq)

        if firstgood == 0:
            raise ValueError("no primes of good reduction in that range")
        else:
            return sorted(periods)

    def _preperiodic_points_to_cyclegraph(self, preper):
        r"""
        Given the complete set of periodic or preperiodic points return the
        digraph representing the orbit.

        If ``preper`` is not the complete set, this function will not fill
        in the gaps.

        INPUT:

        - ``preper`` -- a list or tuple of projective points; the complete
          set of rational periodic or preperiodic points

        OUTPUT:

        A digraph representing the orbit the rational preperiodic points
        ``preper`` in projective space.

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2-2*y^2, y^2])
            sage: preper = [P(-2, 1), P(1, 0), P(0, 1), P(1, 1), P(2, 1), P(-1, 1)]
            sage: f._preperiodic_points_to_cyclegraph(preper)
            Looped digraph on 6 vertices
        """
        V = []
        E = []
        #We store the points we encounter is a list, D. Each new point is checked to
        #see if it is in that list (which uses ==) so that equal points with different
        #representations only appear once in the graph.
        D = []
        for val in preper:
            try:
                V.append(D[D.index(val)])
            except ValueError:
                D.append(val)
                V.append(val)
            Q = self(val)
            Q.normalize_coordinates()
            try:
                E.append([D[D.index(Q)]])
            except ValueError:
                D.append(Q)
                E.append([Q])
        from sage.graphs.digraph import DiGraph
        g = DiGraph(dict(zip(V, E)), loops=True)
        return(g)

    def is_PGL_minimal(self, prime_list=None):
        r"""
        Check if this dynamical system is a minimal model in
        its conjugacy class.

        See [BM2012]_ and [Mol2015]_ for a description of the algorithm.

        INPUT:

        - ``prime_list`` -- (optional) list of primes to check minimality

        OUTPUT: boolean

        EXAMPLES::

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

        ::

            sage: PS.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([6*x^2+12*x*y+7*y^2, 12*x*y])
            sage: f.is_PGL_minimal()
            False

        ::

            sage: PS.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([6*x^2+12*x*y+7*y^2, y^2])
            sage: f.is_PGL_minimal()
            Traceback (most recent call last):
            ...
            TypeError: affine minimality is only considered for maps not of the form
            f or 1/f for a polynomial f
        """
        if self.base_ring() != QQ and self.base_ring() != ZZ:
            raise NotImplementedError("minimal models only implemented over ZZ or QQ")
        if not self.is_morphism():
            raise TypeError("the function is not a morphism")
        if self.degree() == 1:
            raise NotImplementedError("minimality is only for degree 2 or higher")

        from .endPN_minimal_model import affine_minimal
        return(affine_minimal(self, False , prime_list , True))

    def minimal_model(self, return_transformation=False, prime_list=None):
        r"""
        Determine if this dynamical system is minimal.

        This dynamical system must be defined over the projective line
        over the rationals. In particular, determine if this map is affine
        minimal, which is enough to decide if it is minimal or not.
        See Proposition 2.10 in [BM2012]_.

        REFERENCES:

        - [BM2012]_
        - [Mol2015]_

        INPUT:

        - ``return_transformation`` -- (default: ``False``) boolean; this
          signals a return of the `PGL_2` transformation to conjugate
          this map to the calculated minimal model

        - ``prime_list`` -- (optional) a list of primes, in case one
          only wants to determine minimality at those specific primes

        OUTPUT:

        - a scheme morphism on the projective line which is a minimal model
          of this map

        - a `PGL(2,\QQ)` element which conjugates this map to a minimal model

        EXAMPLES::

            sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([X^2+3*Y^2, X*Y])
            sage: f.minimal_model(return_transformation=True)
            (
            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]
            )

        ::

            sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([7365/2*X^4 + 6282*X^3*Y + 4023*X^2*Y^2 + 1146*X*Y^3 + 245/2*Y^4,
            ....:                                 -12329/2*X^4 - 10506*X^3*Y - 6723*X^2*Y^2 - 1914*X*Y^3 - 409/2*Y^4])
            sage: f.minimal_model(return_transformation=True)
            (
            Dynamical System of Projective Space of dimension 1 over Rational
            Field
              Defn: Defined on coordinates by sending (X : Y) to
                    (22176*X^4 + 151956*X^3*Y + 390474*X^2*Y^2 + 445956*X*Y^3 + 190999*Y^4
                     : -12329*X^4 - 84480*X^3*Y - 217080*X^2*Y^2 - 247920*X*Y^3 - 106180*Y^4),
            [2 3]
            [0 1]
            )

        ::

            sage: PS.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([6*x^2+12*x*y+7*y^2, 12*x*y])
            sage: f.minimal_model()
            Dynamical System of Projective Space of dimension 1 over Rational
            Field
              Defn: Defined on coordinates by sending (x : y) to
                    (x^2 + 12*x*y + 42*y^2 : 2*x*y)

        ::

            sage: PS.<x,y> = ProjectiveSpace(ZZ,1)
            sage: f = DynamicalSystem_projective([6*x^2+12*x*y+7*y^2, 12*x*y + 42*y^2])
            sage: g,M=f.minimal_model(return_transformation=True)
            sage: f.conjugate(M) == g
            True

        ::

            sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([X+Y, X-3*Y])
            sage: f.minimal_model()
            Traceback (most recent call last):
            ...
            NotImplementedError: minimality is only for degree 2 or higher

        ::

            sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([X^2-Y^2, X^2+X*Y])
            sage: f.minimal_model()
            Traceback (most recent call last):
            ...
            TypeError: the function is not a morphism

        """
        if self.base_ring() != ZZ and self.base_ring() != QQ:
            raise NotImplementedError("minimal models only implemented over ZZ or QQ")
        if not self.is_morphism():
            raise TypeError("the function is not a morphism")
        if self.degree() == 1:
            raise NotImplementedError("minimality is only for degree 2 or higher")

        from .endPN_minimal_model import affine_minimal
        return(affine_minimal(self, return_transformation, prime_list, False))

    def automorphism_group(self, **kwds):
        r"""
        Calculates the subgroup of `PGL2` that is the automorphism group
        of this dynamical system.

        The automorphism group is the set of `PGL(2)` elements that fixes
        this map under conjugation.

        INPUT:

        keywords:

        - ``starting_prime`` -- (default: 5) the first prime to use for CRT

        - ``algorithm``-- (optional) can be one of the following:

          * ``'CRT'`` - Chinese Remainder Theorem
          * ``'fixed_points'`` - fixed points algorithm

        - ``return_functions``-- (default: ``False``) boolean; ``True``
          returns elements as linear fractional transformations and
          ``False`` returns elements as `PGL2` matrices

        - ``iso_type`` -- (default: ``False``) boolean; ``True`` returns the
          isomorphism type of the automorphism group

        OUTPUT: a list of elements in the automorphism group

        AUTHORS:

        - Original algorithm written by Xander Faber, Michelle Manes,
          Bianca Viray

        - Modified by Joao Alberto de Faria, Ben Hutz, Bianca Thompson

        REFERENCES:

        - [FMV2014]_

        EXAMPLES::

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

        ::

            sage: R.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + 5*x*y + 5*y^2, 5*x^2 + 5*x*y + y^2])
            sage: f.automorphism_group()
            [
            [1 0]  [0 2]
            [0 1], [2 0]
            ]

        ::

            sage: R.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2-2*x*y-2*y^2, -2*x^2-2*x*y+y^2])
            sage: f.automorphism_group(return_functions=True)
            [x, 2/(2*x), -x - 1, -2*x/(2*x + 2), (-x - 1)/x, -1/(x + 1)]

        ::

            sage: R.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([3*x^2*y - y^3, x^3 - 3*x*y^2])
            sage: f.automorphism_group(algorithm='CRT', return_functions=True, iso_type=True)
            ([x, (x + 1)/(x - 1), (-x + 1)/(x + 1), -x, 1/x, -1/x,
            (x - 1)/(x + 1), (-x - 1)/(x - 1)], 'Dihedral of order 8')

        ::

            sage: A.<z> = AffineSpace(QQ,1)
            sage: f = DynamicalSystem_affine([1/z^3])
            sage: F = f.homogenize(1)
            sage: F.automorphism_group()
            [
            [1 0]  [0 2]  [-1  0]  [ 0 -2]
            [0 1], [2 0], [ 0  1], [ 2  0]
            ]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x**2 + x*z, y**2, z**2])
            sage: f.automorphism_group() # long time
            [
            [1 0 0]
            [0 1 0]
            [0 0 1]
            ]

        ::

            sage: K.<w> = CyclotomicField(3)
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: D6 = DynamicalSystem_projective([y^2,x^2])
            sage: D6.automorphism_group()
            [
            [1 0]  [0 w]  [0 1]  [w 0]  [-w - 1      0]  [     0 -w - 1]
            [0 1], [1 0], [1 0], [0 1], [     0      1], [     1      0]
            ]
        """
        alg = kwds.get('algorithm', None)
        p = kwds.get('starting_prime', 5)
        return_functions = kwds.get('return_functions', False)
        iso_type = kwds.get('iso_type', False)
        if self.domain().dimension_relative() != 1:
            return self.conjugating_set(self)
        if self.base_ring() != QQ  and self.base_ring != ZZ:
            return self.conjugating_set(self)
        f = self.dehomogenize(1)
        R = PolynomialRing(f.base_ring(),'x')
        if is_FractionFieldElement(f[0]):
            F = (f[0].numerator().univariate_polynomial(R))/f[0].denominator().univariate_polynomial(R)
        else:
            F = f[0].univariate_polynomial(R)
        from .endPN_automorphism_group import automorphism_group_QQ_CRT, automorphism_group_QQ_fixedpoints
        if alg is None:
            if self.degree() <= 12:
                return(automorphism_group_QQ_fixedpoints(F, return_functions, iso_type))
            return(automorphism_group_QQ_CRT(F, p, return_functions, iso_type))
        elif alg == 'CRT':
            return(automorphism_group_QQ_CRT(F, p, return_functions, iso_type))
        return(automorphism_group_QQ_fixedpoints(F, return_functions, iso_type))

    def critical_subscheme(self):
        r"""
        Return the critical subscheme of this dynamical system.

        OUTPUT: projective subscheme

        EXAMPLES::

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

        ::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([2*x^2-y^2, x*y, z^2])
            sage: f.critical_subscheme()
            Closed subscheme of Projective Space of dimension 2 over Rational Field
            defined by:
            8*x^2*z + 4*y^2*z

        ::

            sage: P.<x,y,z,w> = ProjectiveSpace(GF(81),3)
            sage: g = DynamicalSystem_projective([x^3+y^3, y^3+z^3, z^3+x^3, w^3])
            sage: g.critical_subscheme()
            Closed subscheme of Projective Space of dimension 3 over Finite Field in
            z4 of size 3^4 defined by:
              0

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2,x*y])
            sage: f.critical_subscheme()
            Traceback (most recent call last):
            ...
            TypeError: the function is not a morphism
        """
        PS = self.domain()
        if not is_ProjectiveSpace(PS):
            raise NotImplementedError("not implemented for subschemes")
        if not self.is_morphism():
            raise TypeError("the function is not a morphism")
        wr = self.wronskian_ideal()
        crit_subscheme = self.codomain().subscheme(wr)
        return crit_subscheme

    def critical_points(self, R=None):
        r"""
        Return the critical points of this dynamcial system defined over
        the ring ``R`` or the base ring of this map.

        Must be dimension 1.

        INPUT:

        - ``R`` -- (optional) a ring

        OUTPUT: a list of projective space points defined over ``R``

        EXAMPLES::

            sage: set_verbose(None)
            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^3-2*x*y^2 + 2*y^3, y^3])
            sage: f.critical_points()
            [(1 : 0)]
            sage: K.<w> = QuadraticField(6)
            sage: f.critical_points(K)
            [(-1/3*w : 1), (1/3*w : 1), (1 : 0)]

        ::

            sage: set_verbose(None)
            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([2*x^2-y^2, x*y])
            sage: f.critical_points(QQbar)
            [(-0.7071067811865475?*I : 1), (0.7071067811865475?*I : 1)]
        """
        PS = self.domain()
        if PS.dimension_relative() > 1:
            raise NotImplementedError("use .wronskian_ideal() for dimension > 1")
        if R is None:
            F = self
        else:
            F = self.change_ring(R)
        P = F.codomain()
        X = F.critical_subscheme()
        crit_points = [P(Q) for Q in X.rational_points()]
        return crit_points

    def is_postcritically_finite(self, err=0.01, embedding=None):
        r"""
        Determine if this dynamical system is post-critically finite.

        Only for endomorphisms of `\mathbb{P}^1`. It checks if each critical
        point is preperiodic. The optional parameter ``err`` is passed into
        ``is_preperiodic()`` as part of the preperiodic check.

        INPUT:

        - ``err`` -- (default: 0.01) positive real number

        - ``embedding`` -- embedding of base ring into `\QQbar`

        OUTPUT: boolean

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^3- y^3, y^3])
            sage: f.is_postcritically_finite()
            False

        ::

            sage: R.<z> = QQ[]
            sage: K.<v> = NumberField(z^8 + 3*z^6 + 3*z^4 + z^2 + 1)
            sage: PS.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^3+v*y^3, y^3])
            sage: f.is_postcritically_finite(embedding=K.embeddings(QQbar)[0]) # long time
            True

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([6*x^2+16*x*y+16*y^2, -3*x^2-4*x*y-4*y^2])
            sage: f.is_postcritically_finite()
            True
        """
        #iteration of subschemes not yet implemented
        if self.domain().dimension_relative() > 1:
            raise NotImplementedError("only implemented in dimension 1")

        #Since is_preperiodic uses heights we need to be over a numberfield
        K = FractionField(self.codomain().base_ring())
        if not K in NumberFields() and not K is QQbar:
            raise NotImplementedError("must be over a number field or a number field order or QQbar")

        if embedding is None:
            F = self.change_ring(QQbar)
        else:
            F = self.change_ring(embedding)
        crit_points = F.critical_points()
        pcf = True
        i = 0
        while pcf and i < len(crit_points):
            if crit_points[i].is_preperiodic(F, err) == False:
                pcf = False
            i += 1
        return(pcf)

    def critical_point_portrait(self, check=True, embedding=None):
        r"""
        If this dynamical system  is post-critically finite, return its
        critical point portrait.

        This is the directed graph of iterates starting with the critical
        points. Must be dimension 1. If ``check`` is ``True``, then the
        map is first checked to see if it is postcritically finite.

        INPUT:

        - ``check`` -- boolean

        - ``embedding`` -- embedding of base ring into `\QQbar`

        OUTPUT: a digraph

        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: f = DynamicalSystem_projective([x^2+v*y^2, y^2])
            sage: f.critical_point_portrait(check=False, embedding=K.embeddings(QQbar)[0]) # long time
            Looped digraph on 6 vertices

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^5 + 5/4*x*y^4, y^5])
            sage: f.critical_point_portrait(check=False)
            Looped digraph on 5 vertices

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + 2*y^2, y^2])
            sage: f.critical_point_portrait()
            Traceback (most recent call last):
            ...
            TypeError: map must be post-critically finite
        """
        #input checking done in is_postcritically_finite
        if check:
            if not self.is_postcritically_finite():
                raise TypeError("map must be post-critically finite")
        if embedding is None:
            F = self.change_ring(QQbar)
        else:
            F = self.change_ring(embedding)
        crit_points = F.critical_points()
        N = len(crit_points)
        for i in range(N):
            done = False
            Q= F(crit_points[i])
            while not done:
                if Q in crit_points:
                    done = True
                else:
                    crit_points.append(Q)
                Q = F(Q)
        return(F._preperiodic_points_to_cyclegraph(crit_points))

    def critical_height(self, **kwds):
        r"""
        Compute the critical height of this dynamical system.

        The critical height is defined by J. Silverman as
        the sum of the canonical heights of the critical points.
        This must be dimension 1 and defined over a number field
        or number field order.

        INPUT:

        kwds:

        - ``badprimes`` -- (optional) a list of primes of bad reduction

        - ``N`` -- (default: 10) positive integer; number of terms of
          the series to use in the local green functions

        - ``prec`` -- (default: 100) positive integer, float point
          or `p`-adic precision

        - ``error_bound`` -- (optional) a positive real number

        - ``embedding`` -- (optional) the embedding of the base field to `\QQbar`

        OUTPUT: real number

        EXAMPLES::

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

        ::

            sage: K.<w> = QuadraticField(2)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2+w*y^2, y^2])
            sage: f.critical_height()
            0.16090842452312941163719755472

        Postcritically finite maps have critical height 0::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^3-3/4*x*y^2 + 3/4*y^3, y^3])
            sage: f.critical_height(error_bound=0.0001)
            0.00000000000000000000000000000
        """
        PS = self.codomain()
        if PS.dimension_relative() > 1:
            raise NotImplementedError("only implemented in dimension 1")

        K = FractionField(PS.base_ring())
        if not K in NumberFields() and not K is QQbar:
            raise NotImplementedError("must be over a number field or a number field order or QQbar")
        #doesn't really matter which we choose as Galois conjugates have the same height
        emb = kwds.get("embedding", K.embeddings(QQbar)[0])
        F = self.change_ring(K).change_ring(emb)
        crit_points = F.critical_points()
        n = len(crit_points)
        err_bound = kwds.get("error_bound", None)
        if not err_bound is None:
            kwds["error_bound"] = err_bound / n
        ch = 0
        for P in crit_points:
            ch += F.canonical_height(P, **kwds)
        return ch

    def periodic_points(self, n, minimal=True, R=None, algorithm='variety',
                        return_scheme=False):
        r"""
        Computes the periodic points of period ``n`` of this dynamical system
        defined over the ring ``R`` or the base ring of the map.

        This can be done either by finding the rational points on the variety
        defining the points of period ``n``, or, for finite fields,
        finding the cycle of appropriate length in the cyclegraph. For small
        cardinality fields, the cyclegraph algorithm is effective for any
        map and length cycle, but is slow when the cyclegraph is large.
        The variety algorithm is good for small period, degree, and dimension,
        but is slow as the defining equations of the variety get more
        complicated.

        For rational map, where there are potentially infinitely many peiodic
        points of a given period, you must use the ``return_scheme`` option.
        Note that this scheme will include the indeterminacy locus.

        INPUT:

        - ``n`` - a positive integer

        - ``minimal`` -- (default: ``True``) boolean; ``True`` specifies to
          find only the periodic points of minimal period ``n`` and ``False``
          specifies to find all periodic points of period ``n``

        - ``R`` a commutative ring

        - ``algorithm`` -- (default: ``'variety'``) must be one of
          the following:

          * ``'variety'`` - find the rational points on the appropriate variety
          * ``'cyclegraph'`` - find the cycles from the cycle graph

        - ``return_scheme`` -- return a subscheme of the ambient space
          that defines the ``n`` th periodic points

        OUTPUT:

        A list of periodic points of this map or the subscheme defining
        the periodic points.

        EXAMPLES::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QuadraticField(5,'t'),2)
            sage: f = DynamicalSystem_projective([x^2 - 21/16*z^2, y^2-z^2, z^2])
            sage: f.periodic_points(2)
            [(-5/4 : -1 : 1), (-5/4 : -1/2*t + 1/2 : 1), (-5/4 : 0 : 1),
             (-5/4 : 1/2*t + 1/2 : 1), (-3/4 : -1 : 1), (-3/4 : 0 : 1),
             (1/4 : -1 : 1), (1/4 : -1/2*t + 1/2 : 1), (1/4 : 0 : 1),
             (1/4 : 1/2*t + 1/2 : 1), (7/4 : -1 : 1), (7/4 : 0 : 1)]

        ::

            sage: w = QQ['w'].0
            sage: K = NumberField(w^6 - 3*w^5 + 5*w^4 - 5*w^3 + 5*w^2 - 3*w + 1,'s')
            sage: P.<x,y,z> = ProjectiveSpace(K,2)
            sage: f = DynamicalSystem_projective([x^2+z^2, y^2+x^2, z^2+y^2])
            sage: f.periodic_points(1)
            [(-s^5 + 3*s^4 - 5*s^3 + 4*s^2 - 3*s + 1 : s^5 - 2*s^4 + 3*s^3 - 3*s^2 + 4*s - 1 : 1),
             (-2*s^5 + 4*s^4 - 5*s^3 + 3*s^2 - 4*s : -2*s^5 + 5*s^4 - 7*s^3 + 6*s^2 - 7*s + 3 : 1),
             (-s^5 + 3*s^4 - 4*s^3 + 4*s^2 - 4*s + 2 : -s^5 + 2*s^4 - 2*s^3 + s^2 - s : 1),
             (s^5 - 2*s^4 + 3*s^3 - 3*s^2 + 3*s - 1 : -s^5 + 3*s^4 - 5*s^3 + 4*s^2 - 4*s + 2 : 1),
             (2*s^5 - 6*s^4 + 9*s^3 - 8*s^2 + 7*s - 4 : 2*s^5 - 5*s^4 + 7*s^3 - 5*s^2 + 6*s - 2 : 1),
             (1 : 1 : 1),
             (s^5 - 2*s^4 + 2*s^3 + s : s^5 - 3*s^4 + 4*s^3 - 3*s^2 + 2*s - 1 : 1)]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x^2 - 21/16*z^2, y^2-2*z^2, z^2])
            sage: f.periodic_points(2, False)
            [(-5/4 : -1 : 1), (-5/4 : 2 : 1), (-3/4 : -1 : 1),
             (-3/4 : 2 : 1), (0 : 1 : 0), (1/4 : -1 : 1), (1/4 : 2 : 1),
             (1 : 0 : 0), (1 : 1 : 0), (7/4 : -1 : 1), (7/4 : 2 : 1)]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x^2 - 21/16*z^2, y^2-2*z^2, z^2])
            sage: f.periodic_points(2)
            [(-5/4 : -1 : 1), (-5/4 : 2 : 1), (1/4 : -1 : 1), (1/4 : 2 : 1)]

        ::

            sage: set_verbose(None)
            sage: P.<x,y> = ProjectiveSpace(ZZ, 1)
            sage: f = DynamicalSystem_projective([x^2+y^2,y^2])
            sage: f.periodic_points(2, R=QQbar, minimal=False)
            [(-0.500000000000000? - 1.322875655532296?*I : 1),
             (-0.500000000000000? + 1.322875655532296?*I : 1),
             (0.500000000000000? - 0.866025403784439?*I : 1),
             (0.500000000000000? + 0.866025403784439?*I : 1),
             (1 : 0)]

        ::

            sage: P.<x,y> = ProjectiveSpace(GF(307), 1)
            sage: f = DynamicalSystem_projective([x^10+y^10, y^10])
            sage: f.periodic_points(16, minimal=True, algorithm='cyclegraph')
            [(69 : 1), (185 : 1), (120 : 1), (136 : 1), (97 : 1), (183 : 1),
             (170 : 1), (105 : 1), (274 : 1), (275 : 1), (154 : 1), (156 : 1),
             (87 : 1), (95 : 1), (161 : 1), (128 : 1)]

        ::

            sage: P.<x,y> = ProjectiveSpace(GF(13^2,'t'),1)
            sage: f = DynamicalSystem_projective([x^3 + 3*y^3, x^2*y])
            sage: f.periodic_points(30, minimal=True, algorithm='cyclegraph')
            [(t + 3 : 1), (6*t + 6 : 1), (7*t + 1 : 1), (2*t + 8 : 1),
             (3*t + 4 : 1), (10*t + 12 : 1), (8*t + 10 : 1), (5*t + 11 : 1),
             (7*t + 4 : 1), (4*t + 8 : 1), (9*t + 1 : 1), (2*t + 2 : 1),
             (11*t + 9 : 1), (5*t + 7 : 1), (t + 10 : 1), (12*t + 4 : 1),
             (7*t + 12 : 1), (6*t + 8 : 1), (11*t + 10 : 1), (10*t + 7 : 1),
             (3*t + 9 : 1), (5*t + 5 : 1), (8*t + 3 : 1), (6*t + 11 : 1),
             (9*t + 12 : 1), (4*t + 10 : 1), (11*t + 4 : 1), (2*t + 7 : 1),
             (8*t + 12 : 1), (12*t + 11 : 1)]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([3*x^2+5*y^2,y^2])
            sage: f.periodic_points(2, R=GF(3), minimal=False)
            [(2 : 1)]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: f = DynamicalSystem_projective([x^2, x*y, z^2])
            sage: f.periodic_points(1)
            Traceback (most recent call last):
            ...
            TypeError: use return_scheme=True

        ::

            sage: R.<x> = QQ[]
            sage: K.<u> = NumberField(x^2 - x + 3)
            sage: P.<x,y,z> = ProjectiveSpace(K,2)
            sage: X = P.subscheme(2*x-y)
            sage: f = DynamicalSystem_projective([x^2-y^2, 2*(x^2-y^2), y^2-z^2], domain=X)
            sage: f.periodic_points(2)
            [(-1/5*u - 1/5 : -2/5*u - 2/5 : 1), (1/5*u - 2/5 : 2/5*u - 4/5 : 1)]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x^2-y^2, x^2-z^2, y^2-z^2])
            sage: f.periodic_points(1)
            [(-1 : 0 : 1)]
            sage: f.periodic_points(1, return_scheme=True)
            Closed subscheme of Projective Space of dimension 2 over Rational Field
            defined by:
              -x^3 + x^2*y - y^3 + x*z^2,
              -x*y^2 + x^2*z - y^2*z + x*z^2,
              -y^3 + x^2*z + y*z^2 - z^3
            sage: f.periodic_points(2, minimal=True, return_scheme=True)
            Traceback (most recent call last):
            ...
            NotImplementedError: return_subscheme only implemented for minimal=False
        """
        if n <= 0:
            raise ValueError("a positive integer period must be specified")
        if R is None:
            f = self
            R = self.base_ring()
        else:
            f = self.change_ring(R)
        CR = f.coordinate_ring()
        dom = f.domain()
        PS = f.codomain().ambient_space()
        N = PS.dimension_relative() + 1
        if algorithm == 'cyclegraph':
            if R in FiniteFields():
                g = f.cyclegraph()
                points = []
                for cycle in g.all_simple_cycles():
                    m = len(cycle)-1
                    if minimal:
                        if m == n:
                            points = points + cycle[:-1]
                    else:
                        if n % m == 0:
                            points = points + cycle[:-1]
                return(points)
            else:
                raise TypeError("ring must be finite to generate cyclegraph")
        elif algorithm == 'variety':
            F = f.nth_iterate_map(n)
            L = [F[i]*CR.gen(j) - F[j]*CR.gen(i) for i in range(0,N)
                 for j in range(i+1, N)]
            L = [t for t in L if t != 0]
            X = PS.subscheme(L + list(dom.defining_polynomials()))
            if return_scheme:  # this includes the indeterminacy locus points!
                if minimal and n != 1:
                    raise NotImplementedError("return_subscheme only implemented for minimal=False")
                return X
            if X.dimension() == 0:
                if R in NumberFields() or R is QQbar or R in FiniteFields():
                    Z = f.indeterminacy_locus()
                    points = [dom(Q) for Q in X.rational_points()]
                    good_points = []
                    for Q in points:
                        try:
                            Z(list(Q))
                        except TypeError:
                            good_points.append(Q)
                    points = good_points

                    if not minimal:
                        return points
                    else:
                        #we want only the points with minimal period n
                        #so we go through the list and remove any that
                        #have smaller period by checking the iterates
                        for i in range(len(points)-1,-1,-1):
                            # iterate points to check if minimal
                            P = points[i]
                            for j in range(1,n):
                                P = f(P)
                                if P == points[i]:
                                    points.pop(i)
                                    break
                        return points
                else:
                    raise NotImplementedError("ring must a number field or finite field")
            else: #a higher dimensional scheme
                raise TypeError("use return_scheme=True")
        else:
            raise ValueError("algorithm must be either 'variety' or 'cyclegraph'")

    def multiplier_spectra(self, n, formal=False, embedding=None, type='point'):
        r"""
        Computes the ``n`` multiplier spectra of this dynamical system.

        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.

        INPUT:

        - ``n`` -- a positive integer, the period

        - ``formal`` -- (default: ``False``) boolean; ``True`` specifies
          to find the formal ``n`` multiplier spectra of this map and
          ``False`` specifies to find the ``n`` multiplier spectra

        - ``embedding`` -- embedding of the base field into `\QQbar`

        - ``type`` -- (default: ``'point'``) string; either ``'point'``
          or ``'cycle'`` depending on whether you compute one multiplier
          per point or one per cycle

        OUTPUT: a list of `\QQbar` elements

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([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)
            [0, -7198147681176255644585/256, 848446157556848459363/19683, -3323781962860268721722583135/35184372088832,
            529278480109921/256, -4290991994944936653/2097152, 1061953534167447403/19683, -3086380435599991/9,
            82911372672808161930567/8192, -119820502365680843999, 3553497751559301575157261317/8192]

        ::

            sage: set_verbose(None)
            sage: z = QQ['z'].0
            sage: K.<w> = NumberField(z^4 - 4*z^2 + 1,'z')
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2 - w/4*y^2, y^2])
            sage: f.multiplier_spectra(2, formal=False, embedding=K.embeddings(QQbar)[0], type='cycle')
            [0,
             5.931851652578137? + 0.?e-47*I,
             0.0681483474218635? - 1.930649271699173?*I,
             0.0681483474218635? + 1.930649271699173?*I]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 - 3/4*y^2, y^2])
            sage: f.multiplier_spectra(2, formal=False, type='cycle')
            [0, 1, 1, 9]
            sage: f.multiplier_spectra(2, formal=False, type='point')
            [0, 1, 1, 1, 9]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 - 7/4*y^2, y^2])
            sage: f.multiplier_spectra(3, formal=True, type='cycle')
            [1, 1]
            sage: f.multiplier_spectra(3, formal=True, type='point')
            [1, 1, 1, 1, 1, 1]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, x*y])
            sage: f.multiplier_spectra(1)
            [1, 1, 1]
        """
        PS = self.domain()
        n = Integer(n)

        if (n < 1):
            raise ValueError("period must be a positive integer")
        if not is_ProjectiveSpace(PS):
            raise NotImplementedError("not implemented for subschemes")
        if (PS.dimension_relative() > 1):
            raise NotImplementedError("only implemented for dimension 1")
        if not PS.base_ring() in NumberFields() and not PS.base_ring() is QQbar:
            raise NotImplementedError("self must be a map over a number field")

        if embedding is None:
            f = self.change_ring(QQbar)
        else:
            f = self.change_ring(embedding)

        PS = f.domain()

        if not formal:
            G = f.nth_iterate_map(n)
            F = G[0]*PS.gens()[1] - G[1]*PS.gens()[0]
        else:
            # periodic points of formal period n are the roots of the nth dynatomic polynomial
            K = f._number_field_from_algebraics().as_dynamical_system()
            F = K.dynatomic_polynomial(n)
            if K.domain().base_ring() != QQ: # need to coerce F to poly over QQbar. This is only needed if base ring is not QQ
                abspoly = K.domain().base_ring().absolute_polynomial()
                phi = K.domain().base_ring().hom(QQbar.polynomial_root(abspoly,abspoly.any_root(CIF)),QQbar)
                Kx = K.coordinate_ring()
                QQbarx = QQbar[Kx.variable_names()]
                phix = Kx.hom(phi,QQbarx)
                F = phix(F)

        other_roots = F.parent()(F([(f.domain().gens()[0]),1])).univariate_polynomial().roots(ring=QQbar)

        points = []

        minfty = min([e[1] for e in F.exponents()]) # include the point at infinity with the right multiplicity
        for i in range(minfty):
            points.append(PS([1,0]))

        for R in other_roots:
            for i in range(R[1]):
                points.append(PS([R[0],1])) # include copies of higher multiplicity roots

        if type == 'cycle':
            # should include one representative point per cycle, included with the right multiplicity
            newpoints = []

            while points:
                P = points[0]
                newpoints.append(P)
                points.pop(0)
                Q = P
                for i in range(1,n):
                    try:
                        points.remove(f(Q))
                    except ValueError:
                        pass
                    Q = f(Q)
            points = newpoints

        multipliers = [f.multiplier(pt,n)[0,0] for pt in points]

        return multipliers

    def sigma_invariants(self, n, formal=False, embedding=None, type='point'):
        r"""
        Computes the values of the elementary symmetric polynomials of
        the ``n`` multilpier spectra of this dynamical system.

        Can specify to instead compute the values corresponding to the
        elementary symmetric polynomials of the formal ``n`` multiplier
        spectra. The map must be defined over projective space of dimension
        `1`. The base ring should be a number field, number field order, or
        a finite field or a polynomial ring or function field over a
        number field, number field order, or finite field.

        The parameter ``type`` determines if the sigma are computed from
        the multipliers calculated at 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, this is still considered one cycle. In other words, if a
        4-cycle collapses into a 2-cycle with multiplicity 2, there is only
        one multiplier used for the doubled 2-cycle when computing ``n=4``.

        ALGORITHM:

        We use the Poisson product of the resultant of two polynomials:

        .. MATH::

            res(f,g) = \prod_{f(a)=0} g(a).

        Letting `f` be the polynomial defining the periodic or formal
        periodic points and `g` the polynomial `w - f'` for an auxilarly
        variable `w`. Note that if `f` is a rational function, we clear
        denominators for `g`.

        INPUT:

        - ``n`` -- a positive integer, the period

        - ``formal`` -- (default: ``False``) boolean; ``True`` specifies
          to find the values of the elementary symmetric polynomials
          corresponding to the formal ``n`` multiplier spectra and ``False``
          specifies to instead find the values corresponding to the ``n``
          multiplier spectra, which includes the multipliers of all
          periodic points of period ``n``

        - ``embedding`` -- deprecated in :trac:`23333`

        - ``type`` -- (default: ``'point'``) string; either ``'point'``
          or ``'cycle'`` depending on whether you compute with one
          multiplier per point or one per cycle

        OUTPUT: a list of elements in the base ring

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([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)
            [19575526074450617/1048576, -9078122048145044298567432325/2147483648,
            -2622661114909099878224381377917540931367/1099511627776,
            -2622661107937102104196133701280271632423/549755813888,
            338523204830161116503153209450763500631714178825448006778305/72057594037927936, 0]

        ::

            sage: set_verbose(None)
            sage: z = QQ['z'].0
            sage: K = NumberField(z^4 - 4*z^2 + 1, 'z')
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: f = DynamicalSystem_projective([x^2 - 5/4*y^2, y^2])
            sage: f.sigma_invariants(2, formal=False, type='cycle')
            [13, 11, -25, 0]
            sage: f.sigma_invariants(2, formal=False, type='point')
            [12, -2, -36, 25, 0]

        check that infinity as part of a longer cycle is handled correctly::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([y^2, x^2])
            sage: f.sigma_invariants(2, type='cycle')
            [12, 48, 64, 0]
            sage: f.sigma_invariants(2, type='point')
            [12, 48, 64, 0, 0]
            sage: f.sigma_invariants(2, type='cycle', formal=True)
            [0]
            sage: f.sigma_invariants(2, type='point', formal=True)
            [0, 0]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: f = DynamicalSystem_projective([x^2, y^2, z^2])
            sage: f.sigma_invariants(1)
            Traceback (most recent call last):
            ...
            NotImplementedError: only implemented for dimension 1

        ::

            sage: K.<w> = QuadraticField(3)
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: f = DynamicalSystem_projective([x^2 - w*y^2, (1-w)*x*y])
            sage: f.sigma_invariants(2, formal=False, type='cycle')
            [6*w + 21, 78*w + 159, 210*w + 367, 90*w + 156]
            sage: f.sigma_invariants(2, formal=False, type='point')
            [6*w + 24, 96*w + 222, 444*w + 844, 720*w + 1257, 270*w + 468]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + x*y + y^2, y^2 + x*y])
            sage: f.sigma_invariants(1)
            [3, 3, 1]

        ::

            sage: R.<c> = QQ[]
            sage: Pc.<x,y> = ProjectiveSpace(R, 1)
            sage: f = DynamicalSystem_projective([x^2 + c*y^2, y^2])
            sage: f.sigma_invariants(1)
            [2, 4*c, 0]
            sage: f.sigma_invariants(2, formal=True, type='point')
            [8*c + 8, 16*c^2 + 32*c + 16]
            sage: f.sigma_invariants(2, formal=True, type='cycle')
            [4*c + 4]

        doubled fixed point::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^2 - 3/4*y^2, y^2])
            sage: f.sigma_invariants(2, formal=True)
            [2, 1]

        doubled 2 cycle::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^2 - 5/4*y^2, y^2])
            sage: f.sigma_invariants(4, formal=False, type='cycle')
            [170, 5195, 172700, 968615, 1439066, 638125, 0]
        """
        n = ZZ(n)
        if n < 1:
            raise ValueError("period must be a positive integer")
        dom = self.domain()
        if not is_ProjectiveSpace(dom):
            raise NotImplementedError("not implemented for subschemes")
        if dom.dimension_relative() > 1:
            raise NotImplementedError("only implemented for dimension 1")
        if not embedding is None:
            from sage.misc.superseded import deprecation
            deprecation(23333, "embedding keyword no longer used")
        if self.degree() <= 1:
            raise TypeError("must have degree at least 2")
        if not type in ['point', 'cycle']:
            raise ValueError("type must be either point or cycle")

        base_ring = dom.base_ring()
        if is_FractionField(base_ring):
            base_ring = base_ring.ring()
        if (is_PolynomialRing(base_ring) or is_MPolynomialRing(base_ring)
            or base_ring in FunctionFields()):
            base_ring = base_ring.base_ring()
        from sage.rings.number_field.order import is_NumberFieldOrder
        if not (base_ring in NumberFields() or is_NumberFieldOrder(base_ring)
                or (base_ring in FiniteFields())):
            raise NotImplementedError("incompatible base field, see documentation")

        #now we find the two polynomials for the resultant
        Fn = self.nth_iterate_map(n)
        fn = Fn.dehomogenize(1)
        R = fn.domain().coordinate_ring()
        S = PolynomialRing(FractionField(self.base_ring()), 'z', 2)
        phi = R.hom([S.gen(0)], S)
        psi = dom.coordinate_ring().hom([S.gen(0), 1], S)  #dehomogenize
        dfn = fn[0].derivative(R.gen())

        #polynomial to be evaluated at the periodic points
        mult_poly = phi(dfn.denominator())*S.gen(1) - phi(dfn.numerator()) #w-f'(z)

        #polynomial defining the periodic points
        x,y = dom.gens()
        if formal:
            fix_poly = self.dynatomic_polynomial(n)  #f(z)-z
        else:
            fix_poly = Fn[0]*y - Fn[1]*x #f(z) - z

        #check infinity
        inf = dom(1,0)
        inf_per = ZZ(1)
        Q = self(inf)
        while Q != inf and inf_per <= n:
            inf_per += 1
            Q = self(Q)
        #get multiplicity
        if inf_per <= n:
            e_inf = 0
            while (y**(e_inf + 1)).divides(fix_poly):
                e_inf += 1

        if type == 'cycle':
            #now we need to deal with having the correct number of factors
            #1 multiplier for each cycle. But we need to be careful about
            #the length of the cycle and the mutliplicities
            good_res = 1
            if formal:
                #then we are working with the n-th dynatomic and just need
                #to take one multiplier per cycle

                #evaluate the resultant
                fix_poly = psi(fix_poly)
                res = fix_poly.resultant(mult_poly, S.gen(0))
                #take infinty into consideration
                if inf_per.divides(n):
                    res *= (S.gen(1) - self.multiplier(inf, n)[0,0])**e_inf
                res = res.univariate_polynomial()
                #adjust multiplicities
                L = res.factor()
                for p,e in L:
                    good_res *= p**(e/n)
            else:
                #For each d-th dynatomic for d dividing n, take
                #one multiplier per cycle; e.g., this treats a double 2
                #cycle as a single 4 cycle for n=4
                for d in n.divisors():
                    fix_poly_d = self.dynatomic_polynomial(d)
                    resd = mult_poly.resultant(psi(fix_poly_d), S.gen(0))
                    #check infinity
                    if inf_per == d:
                        e_inf_d = 0
                        while (y**(e_inf_d + 1)).divides(fix_poly_d):
                            e_inf_d += 1
                        resd *= (S.gen(1) - self.multiplier(inf, n)[0,0])**e_inf
                    resd = resd.univariate_polynomial()
                    Ld = resd.factor()
                    for pd,ed in Ld:
                        good_res *= pd**(ed/d)
            res = good_res
        else: #type is 'point'
            #evaluate the resultant
            fix_poly = psi(fix_poly)
            res = fix_poly.resultant(mult_poly, S.gen(0))
            #take infinty into consideration
            if inf_per.divides(n):
                res *= (S.gen(1) - self.multiplier(inf, n)[0,0])**e_inf
            res = res.univariate_polynomial()

        #the sigmas are the coeficients
        #need to fix the signs and the order
        sig = res.coefficients(sparse=False)
        den = sig.pop(-1)
        sig.reverse()
        sig = [sig[i] * (-1)**(i+1) / den for i in range(len(sig))]
        return sig

    def reduced_form(self, prec=300, return_conjugation=True, error_limit=0.000001):
        r"""
        Return reduced form of this dynamical system.

        The reduced form is the `SL(2, \ZZ)` equivalent morphism obtained
        by applying the binary form reduction algorithm from Stoll and
        Cremona [SC]_ to the homogeneous polynomial defining the periodic
        points (the dynatomic polynomial). The smallest period `n` with
        enough periodic points is used.

        This should also minimize the sum of the squares of the coefficients,
        but this is not always the case.

        See :meth:`sage.rings.polynomial.multi_polynomial.reduced_form` for
        the information on binary form reduction.

        Implemented by Rebecca Lauren Miller as part of GSOC 2016.

        INPUT:

        - ``prec`` -- (default: 300) integer, desired precision

        - ``return_conjuagtion`` -- (default: ``True``) boolean; return
          an element of `SL(2, \ZZ)`

        - ``error_limit`` -- (default: 0.000001) a real number, sets
          the error tolerance

        OUTPUT:

        - a projective morphism

        - a matrix

        EXAMPLES::

            sage: PS.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^3 + x*y^2, y^3])
            sage: m = matrix(QQ, 2, 2, [-221, -1, 1, 0])
            sage: f = f.conjugate(m)
            sage: f.reduced_form(prec=100) #needs 2 periodic
            Traceback (most recent call last):
            ...
            ValueError: not enough precision
            sage: f.reduced_form()
            (
            Dynamical System of Projective Space of dimension 1 over Rational Field
            Defn: Defined on coordinates by sending (x : y) to
            (x^3 + x*y^2 : y^3)
            ,
            [  0  -1]
            [  1 221]
            )

        ::

            sage: PS.<x,y> = ProjectiveSpace(ZZ, 1)
            sage: f = DynamicalSystem_projective([x^2+ x*y, y^2]) #needs 3 periodic
            sage: m = matrix(QQ, 2, 2, [-221, -1, 1, 0])
            sage: f = f.conjugate(m)
            sage: f.reduced_form(prec=200)
            (
            Dynamical System of Projective Space of dimension 1 over Integer Ring
            Defn: Defined on coordinates by sending (x : y) to
            (-x^2 + x*y - y^2 : -y^2)
            ,
            [  0  -1]
            [  1 220]
            )

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^3, y^3])
            sage: f.reduced_form()
            (
            Dynamical System of Projective Space of dimension 1 over Rational Field
              Defn: Defined on coordinates by sending (x : y) to
                    (x^3 : y^3)
            ,
            <BLANKLINE>
            [-1  0]
            [ 0 -1]
            )

        ::

            sage: PS.<X,Y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([7365*X^4 + 12564*X^3*Y + 8046*X^2*Y^2 + 2292*X*Y^3 + 245*Y^4,\
            -12329*X^4 - 21012*X^3*Y - 13446*X^2*Y^2 - 3828*X*Y^3 - 409*Y^4])
            sage: f.reduced_form(prec=30)
            Traceback (most recent call last):
            ...
            ValueError: accuracy of Newton's root not within tolerance(1.2519607 > 1e-06), increase precision
            sage: f.reduced_form()
            (
            Dynamical System of Projective Space of dimension 1 over Rational Field
              Defn: Defined on coordinates by sending (X : Y) to
                    (-7*X^4 - 12*X^3*Y - 42*X^2*Y^2 - 12*X*Y^3 - 7*Y^4 : -X^4 - 4*X^3*Y - 6*X^2*Y^2 - 4*X*Y^3 - Y^4),
            <BLANKLINE>
            [-1  2]
            [ 2 -5]
            )

        ::

            sage: P.<x,y> = ProjectiveSpace(RR, 1)
            sage: f = DynamicalSystem_projective([x^4, RR(sqrt(2))*y^4])
            sage: m = matrix(RR, 2, 2, [1,12,0,1])
            sage: f = f.conjugate(m)
            sage: f.reduced_form()
            (
            Dynamical System of Projective Space of dimension 1 over Real Field with 53 bits of precision
              Defn: Defined on coordinates by sending (x : y) to
                    (-x^4 + 2.86348722511320e-12*y^4 : -1.41421356237310*y^4)
            ,
            <BLANKLINE>
            [-1 12]
            [ 0 -1]
            )

        ::

            sage: P.<x,y> = ProjectiveSpace(CC, 1)
            sage: f = DynamicalSystem_projective([x^4, CC(sqrt(-2))*y^4])
            sage: m = matrix(CC, 2, 2, [1,12,0,1])
            sage: f = f.conjugate(m)
            sage: f.reduced_form()
            (
            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
                    (-x^4 + (-1.03914726748259e-15)*y^4 : (-8.65956056235493e-17 - 1.41421356237309*I)*y^4) ,
            <BLANKLINE>
            [-1 12]
            [ 0 -1]
            )

        ::

            sage: K.<w> = QuadraticField(2)
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: f = DynamicalSystem_projective([x^3, w*y^3])
            sage: m = matrix(K, 2, 2, [1,12,0,1])
            sage: f = f.conjugate(m)
            sage: f.reduced_form()
            (
            Dynamical System of Projective Space of dimension 1 over Number Field in w with defining polynomial x^2 - 2
              Defn: Defined on coordinates by sending (x : y) to
                    (x^3 : (w)*y^3)
            ,
            <BLANKLINE>
            [-1 12]
            [ 0 -1]
            )

        ::

            sage: R.<x> = QQ[]
            sage: K.<w> = NumberField(x^5+x-3, embedding=(x^5+x-3).roots(ring=CC)[0][0])
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: f = DynamicalSystem_projective([12*x^3, 2334*w*y^3])
            sage: m = matrix(K, 2, 2, [-12,1,1,0])
            sage: f = f.conjugate(m)
            sage: f.reduced_form()
            (
            Dynamical System of Projective Space of dimension 1 over Number Field
            in w with defining polynomial x^5 + x - 3
              Defn: Defined on coordinates by sending (x : y) to
                    (12*x^3 : (2334*w)*y^3)
            ,
            [  0  -1]
            [  1 -12]
            )
        """
        R = self.coordinate_ring()
        F = R(self.dynatomic_polynomial(1))
        x,y = R.gens()
        d = gcd(F, F.derivative(x)).degree() #counts multiple roots
        n = 2
        # Checks to make sure there are enough distinct, roots we need 3
        # if there are not it finds the nth periodic points until there are enough
        while F.degree() - d <= 2:
            F = self.dynatomic_polynomial(n) # finds n periodic points
            d = gcd(F, F.derivative(x)).degree() #counts multiple roots
            n += 1
        G,m = F.reduced_form(prec=prec, return_conjugation=return_conjugation)
        if return_conjugation:
            return (self.conjugate(m), m)
        return self.conjugate(m)

    def _is_preperiodic(self, P, err=0.1, return_period=False):
        r"""
        Determine if the point is preperiodic with respect to this
        dynamical system.

        .. NOTE::

            This is only implemented for projective space (not subschemes).

        ALGORITHM:

        We know that a point is preperiodic if and only if it has
        canonical height zero. However, we can only compute the canonical
        height up to numerical precision. This function first computes
        the canonical height of the point to the given error bound. If
        it is larger than that error bound, then it must not be preperiodic.
        If it is less than the error bound, then we expect preperiodic. In
        this case we begin computing the orbit stopping if either we
        determine the orbit is finite, or the height of the point is large
        enough that it must be wandering. We can determine the height
        cutoff by computing the height difference constant, i.e., the bound
        between the height and the canonical height of a point (which
        depends only on the map and not the point itself). If the height
        of the point is larger than the difference bound, then the canonical
        height cannot be zero so the point cannot be preperiodic.

        INPUT:

        - ``P`` -- a point of this dynamical system's codomain

        kwds:

        - ``error_bound`` -- (default: 0.1) a positive real number;
          sets the error_bound used in the canonical height computation
          and ``return_period`` a boolean which

        - ``return_period`` -- (default: ``False``) boolean; controls if
          the period is returned if the point is preperiodic

        OUTPUT:

        - boolean -- ``True`` if preperiodic

        - if ``return_period`` is ``True``, then ``(0,0)`` if wandering,
          and ``(m,n)`` if preperiod ``m`` and period ``n``

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^3-3*x*y^2, y^3], domain=P)
            sage: Q = P(-1, 1)
            sage: f._is_preperiodic(Q)
            True

        Check that :trac:`23814` is fixed (works even if domain is not specified)::

            sage: R.<X> = PolynomialRing(QQ)
            sage: K.<a> = NumberField(X^2 + X - 1)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2-2*y^2, y^2])
            sage: Q = P.point([a,1])
            sage: Q.is_preperiodic(f)
            True
        """
        if not is_ProjectiveSpace(self.codomain()):
            raise NotImplementedError("must be over projective space")
        if not self.is_morphism():
            raise TypeError("must be a morphism")
        if not P.codomain() == self.domain():
            raise TypeError("point must be in domain of map")

        K = FractionField(self.codomain().base_ring())
        if not K in NumberFields() and not K is QQbar:
            raise NotImplementedError("must be over a number field or"
                                      " a number field order or QQbar")

        h = self.canonical_height(P, error_bound = err)
        # we know canonical height 0 if and only if preperiodic
        # however precision issues can occur so we can only tell *not* preperiodic
        # if the value is larger than the error
        if h <= err:
            # if the canonical height is less than than the
            # error, then we suspect preperiodic so check
            # either we can find the cycle or the height is
            # larger than the difference between the canonical height
            # and the height, so the canonical height cannot be 0
            B = self.height_difference_bound()
            orbit = [P]
            n = 1 # to compute period
            Q = self(P)
            H = Q.global_height()
            while Q not in orbit and H <= B:
                orbit.append(Q)
                Q = self(Q)
                H = Q.global_height()
                n += 1
            if H <= B: #it must have been in the cycle
                if return_period:
                    m = orbit.index(Q)
                    return((m, n-m))
                else:
                    return True
        if return_period:
            return (0,0)
        else:
            return False

class DynamicalSystem_projective_field(DynamicalSystem_projective,
                                       SchemeMorphism_polynomial_projective_space_field):

    def lift_to_rational_periodic(self, points_modp, B=None):
        r"""
        Given a list of points in projective space over `\GF{p}`,
        determine if they lift to `\QQ`-rational periodic points.

        The map must be an endomorphism of projective space defined
        over `\QQ`.

        ALGORITHM:

        Use Hensel lifting to find a `p`-adic approximation for that
        rational point. The accuracy needed is determined by the height
        bound ``B``. Then apply the LLL algorithm to determine if the
        lift corresponds to a rational point.

        If the point is a point of high multiplicity (multiplier 1), the
        procedure can be very slow.

        INPUT:

        - ``points_modp`` -- a list or tuple of pairs containing a point
          in projective space over `\GF{p}` and the possible period

        - ``B`` -- (optional) a positive integer; the height bound for
          a rational preperiodic point

        OUTPUT: a list of projective points

        EXAMPLES::

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

        ::

            There may be multiple points in the lift.
            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([-5*x^2 + 4*y^2, 4*x*y])
            sage: f.lift_to_rational_periodic([[P(1,0).change_ring(GF(3)), 1]]) # long time
            [[(1 : 0), 1], [(2/3 : 1), 1], [(-2/3 : 1), 1]]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([16*x^2 - 29*y^2, 16*y^2])
            sage: f.lift_to_rational_periodic([[P(3,1).change_ring(GF(13)), 3]])
            [[(-1/4 : 1), 3]]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: f = DynamicalSystem_projective([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.lift_to_rational_periodic([[P(14,19,1).change_ring(GF(23)), 9]]) # long time
            [[(-9 : -4 : 1), 9]]
        """
        if not points_modp:
            return []

        if B is None:
            B = e ** self.height_difference_bound()

        p = points_modp[0][0].codomain().base_ring().characteristic()
        if p == 0:
            raise TypeError("must be positive characteristic")
        PS = self.domain()
        N = PS.dimension_relative()
        R = RealField()
        #compute the maximum p-adic precision needed to conclusively determine
        #if the rational point exists
        L = R((R(2 ** (N/2 + 1) * sqrt(N+1) * B**2).log()) / R(p).log() + 1).trunc()

        points = []
        for i in range(len(points_modp)):
            #[point mod p, period, current p-adic precision]
            points.append([points_modp[i][0].change_ring(QQ, check=False), points_modp[i][1], 1])
        good_points = []
        #shifts is used in non-Hensel lifting
        shifts = None
        #While there are still points to consider try to lift to next precision
        while points:
            q = points.pop()
            qindex = N
            #Find the last non-zero coordinate to use for normalizations
            while q[0][qindex] % p == 0:
                qindex -= 1
            T = q[0]
            n = q[1]
            k = q[2]
            T.scale_by(1 / T[qindex]) #normalize
            bad = 0
            #stop where we reach the needed precision or the point is bad
            while k < L and bad == 0:
                l = self._multipliermod(T, n, p, 2*k)
                l -= l.parent().one() #f^n(x) - x
                lp = l.change_ring(Zmod(p**k))
                ldet = lp.determinant()
                # if the matrix is invertible then we can Hensel lift
                if ldet % p != 0:
                    RQ = ZZ.quo(p**(2*k))
                    T.clear_denominators()
                    newT = T.change_ring(RQ, check=False)
                    fp = self.change_ring(RQ, check=False)
                    S = fp.nth_iterate(newT, n, normalize=False).change_ring(QQ, check=False)
                    T.scale_by(1 / T[qindex])
                    S.scale_by(1 / S[qindex])
                    newS = list(S)
                    for i in range(N + 1):
                        newS[i] = S[i] - T[i]
                        if newS[i] % (p**k) != 0 and i != N:
                            bad = 1
                            break
                    if bad == 1:
                        break
                    S = PS.point(newS, False)
                    S.scale_by(-1 / p**k)
                    vecs = [Zmod(p**k)(S._coords[iS]) for iS in range(N + 1)]
                    vecs.pop(qindex)
                    newvecs = list((lp.inverse()) * vector(vecs)) #l.inverse should be mod p^k!!
                    newS = []
                    [newS.append(QQ(newvecs[i])) for i in range(qindex)]
                    newS.append(0)
                    [newS.append(QQ(newvecs[i])) for i in range(qindex, N)]
                    for i in range(N + 1):
                        newS[i] = newS[i] % (p**k)
                    S = PS.point(newS, False) #don't check for [0,...,0]
                    newT = list(T)
                    for i in range(N + 1):
                        newT[i] += S[i] * (p**k)
                    T = PS.point(newT, False)
                    T.normalize_coordinates()
                    #Hensel gives us 2k for the newprecision
                    k = min(2*k, L)
                else:
                    #we are unable to Hensel Lift so must try all possible lifts
                    #to the next precision (k+1)
                    first = 0
                    newq = []
                    RQ = Zmod(p**(k+1))
                    fp = self.change_ring(RQ, check=False)
                    if shifts is None:
                        shifts = xmrange([p for i in range(N)])
                    for shift in shifts:
                        newT = [RQ(t) for t in T]  #T.change_ring(RQ, check = False)
                        shiftindex = 0
                        for i in range(N + 1):
                            if i != qindex:
                                newT[i] = newT[i] + shift[shiftindex] * p**k
                                shiftindex += 1
                        newT = fp.domain().point(newT, check=False)
                        TT = fp.nth_iterate(newT, n, normalize=False)
                        if TT == newT:
                            if first == 0:
                                newq.append(newT.change_ring(QQ, check=False))
                                newq.append(n)
                                newq.append(k + 1)
                                first = 1
                            else:
                                points.append([newT.change_ring(QQ, check=False), n, k+1])
                    if not newq:
                        bad = 1
                        break
                    else:
                        T = newq[0]
                        k += 1
            #given a p-adic lift of appropriate precision
            #perform LLL to find the "smallest" rational approximation
            #If this height is small enough, then it is a valid rational point
            if bad == 0:
                M = matrix(N + 2, N + 1)
                T.clear_denominators()
                for i in range(N + 1):
                    M[0, i] = T[i]
                    M[i+1, i] = p**L
                M[N+1, N] = p**L
                M = M.LLL()
                Q = []
                [Q.append(M[1, i]) for i in range(N + 1)]
                g = gcd(Q)
                #remove gcds since this is a projective point
                newB = B * g
                for i in range(N + 1):
                    if abs(Q[i]) > newB:
                        #height too big, so not a valid point
                        bad = 1
                        break
                if bad == 0:
                    P = PS.point(Q, False)
                    #check that it is actually periodic
                    newP = copy(P)
                    k = 1
                    done = False
                    while not done and k <= n:
                          newP = self(newP)
                          if newP == P:
                              if not ([P, k] in good_points):
                                  good_points.append([newP, k])
                              done = True
                          k += 1

        return(good_points)

    def rational_periodic_points(self, **kwds):
        r"""
        Determine the set of rational periodic points
        for this dynamical system.

        The map must be defined over `\QQ` and be an endomorphism of
        projective space. If the map is a polynomial endomorphism of
        `\mathbb{P}^1`, i.e. has a totally ramified fixed point, then
        the base ring can be an absolute number field.
        This is done by passing to the Weil restriction.

        The default parameter values are typically good choices for
        `\mathbb{P}^1`. If you are having trouble getting a particular
        map to finish, try first computing the possible periods, then
        try various different ``lifting_prime`` values.

        ALGORITHM:

        Modulo each prime of good reduction `p` determine the set of
        periodic points modulo `p`. For each cycle modulo `p` compute
        the set of possible periods (`mrp^e`). Take the intersection
        of the list of possible periods modulo several primes of good
        reduction to get a possible list of minimal periods of rational
        periodic points. Take each point modulo `p` associated to each
        of these possible periods and try to lift it to a rational point
        with a combination of `p`-adic approximation and the LLL basis
        reduction algorithm.

        See [Hutz2015]_.

        INPUT:

        kwds:

        - ``prime_bound`` -- (default: ``[1,20]``) a pair (list or tuple)
          of positive integers that represent the limits of primes to use
          in the reduction step or an integer that represents the upper bound

        - ``lifting_prime`` -- (default: 23) a prime integer; argument that
          specifies modulo which prime to try and perform the lifting

        - ``periods`` -- (optional) a list of positive integers that is
          the list of possible periods

        - ``bad_primes`` -- (optional) a list or tuple of integer primes;
          the primes of bad reduction

        - ``ncpus`` -- (default: all cpus) number of cpus to use in parallel

        OUTPUT: a list of rational points in projective space

        EXAMPLES::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([2*x^3 - 50*x*z^2 + 24*z^3,
            ....:                                 5*y^3 - 53*y*z^2 + 24*z^3, 24*z^3])
            sage: sorted(f.rational_periodic_points(prime_bound=[1,20])) # long time
            [(-3 : -1 : 1), (-3 : 0 : 1), (-3 : 1 : 1), (-3 : 3 : 1), (-1 : -1 : 1),
             (-1 : 0 : 1), (-1 : 1 : 1), (-1 : 3 : 1), (0 : 1 : 0), (1 : -1 : 1),
             (1 : 0 : 0), (1 : 0 : 1), (1 : 1 : 1), (1 : 3 : 1), (3 : -1 : 1),
             (3 : 0 : 1), (3 : 1 : 1), (3 : 3 : 1), (5 : -1 : 1), (5 : 0 : 1),
             (5 : 1 : 1), (5 : 3 : 1)]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([-5*x^2 + 4*y^2, 4*x*y])
            sage: sorted(f.rational_periodic_points()) # long time
            [(-2 : 1), (-2/3 : 1), (2/3 : 1), (1 : 0), (2 : 1)]

        ::

            sage: R.<x> = QQ[]
            sage: K.<w> = NumberField(x^2-x+1)
            sage: P.<u,v> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([u^2 + v^2,v^2])
            sage: f.rational_periodic_points()
            [(w : 1), (1 : 0), (-w + 1 : 1)]

        ::

            sage: R.<x> = QQ[]
            sage: K.<w> = NumberField(x^2-x+1)
            sage: P.<u,v> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([u^2+v^2,u*v])
            sage: f.rational_periodic_points()
            Traceback (most recent call last):
            ...
            NotImplementedError: rational periodic points for number fields only implemented for polynomials
        """
        PS = self.domain()
        K = PS.base_ring()
        if K in NumberFields():
            if not K.is_absolute():
                raise TypeError("base field must be an absolute field")
            d = K.absolute_degree()
            #check that we are not over QQ
            if d > 1:
                if PS.dimension_relative() != 1:
                    raise NotImplementedError("rational periodic points for number fields only implemented in dimension 1")
                w = K.absolute_generator()
                #we need to dehomogenize for the Weil restriction and will check that point at infty
                #separately. We also check here that we are working with a polynomial. If the map
                #is not a polynomial, the Weil restriction will not be a morphism and we cannot
                #apply this algorithm.
                g = self.dehomogenize(1)
                inf = PS([1,0])
                k = 1
                if isinstance(g[0], FractionFieldElement):
                    g = self.dehomogenize(0)
                    inf = PS([0,1])
                    k = 0
                    if isinstance(g[0], FractionFieldElement):
                        raise NotImplementedError("rational periodic points for number fields only implemented for polynomials")
                #determine rational periodic points
                #infinity is a totally ramified fixed point for a polynomial
                periodic_points = set([inf])
                #compute the weil restriction
                G = g.weil_restriction()
                F = G.homogenize(d)
                #find the QQ rational periodic points for the weil restriction
                Fper = F.rational_periodic_points(**kwds)
                for P in Fper:
                    #take the 'good' points in the weil restriction and find the
                    #associated number field points.
                    if P[d] == 1:
                        pt = [sum([P[i]*w**i for i in range(d)])]
                        pt.insert(k,1)
                        Q = PS(pt)
                        #for each periodic point get the entire cycle
                        if not Q in periodic_points:
                            #check periodic not preperiodic and add all points in cycle
                            orb = set([Q])
                            Q2 = self(Q)
                            while Q2 not in orb:
                                orb.add(Q2)
                                Q2 = self(Q2)
                            if Q2 == Q:
                                periodic_points = periodic_points.union(orb)
                return list(periodic_points)
            else:
                primebound = kwds.pop("prime_bound", [1, 20])
                p = kwds.pop("lifting_prime", 23)
                periods = kwds.pop("periods", None)
                badprimes = kwds.pop("bad_primes", None)
                num_cpus = kwds.pop("ncpus", ncpus())

                if not isinstance(primebound, (list, tuple)):
                    try:
                        primebound = [1, ZZ(primebound)]
                    except TypeError:
                        raise TypeError("bound on primes must be an integer")
                else:
                    try:
                        primebound[0] = ZZ(primebound[0])
                        primebound[1] = ZZ(primebound[1])
                    except TypeError:
                        raise TypeError("prime bounds must be integers")

                if badprimes is None:
                    badprimes = self.primes_of_bad_reduction()
                if periods is None:
                    periods = self.possible_periods(prime_bound=primebound, bad_primes=badprimes, ncpus=num_cpus)
                PS = self.domain()
                periodic = set()
                while p in badprimes:
                    p = next_prime(p + 1)
                B = e ** self.height_difference_bound()

                f = self.change_ring(GF(p))
                all_points = f.possible_periods(True) #return the list of points and their periods.
                pos_points = []
                for i in range(len(all_points)):
                    if all_points[i][1] in periods and not (all_points[i] in pos_points):  #check period, remove duplicates
                        pos_points.append(all_points[i])
                periodic_points = self.lift_to_rational_periodic(pos_points,B)
                for p,n in periodic_points:
                    for k in range(n):
                        p.normalize_coordinates()
                        periodic.add(p)
                        p = self(p)
                return list(periodic)
        else:
            raise TypeError("base field must be an absolute number field")

    def all_rational_preimages(self, points):
        r"""
        Given a set of rational points in the domain of this
        dynamical system, return all the rational preimages of those points.

        In others words, all the rational points which have some
        iterate in the set points. This function repeatedly calls
        ``rational_preimages``. If the degree is at least two,
        by Northocott, this is always a finite set. The map must be
        defined over number fields and be an endomorphism.

        INPUT:

        - ``points`` -- a list of rational points in the domain of this map

        OUTPUT: a list of rational points in the domain of this map

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([16*x^2 - 29*y^2, 16*y^2])
            sage: sorted(f.all_rational_preimages([P(-1,4)]))
            [(-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: f = DynamicalSystem_projective([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: f = DynamicalSystem_projective([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: f = DynamicalSystem_projective([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: f = DynamicalSystem_projective([u^2+v^2, v^2])
            sage: f.all_rational_preimages(P(4))
            [(-w : 1), (w : 1)]
        """
        if self.domain().base_ring() not in NumberFields():
            raise TypeError("field won't return finite list of elements")
        if not isinstance(points, (list, tuple)):
            points = [points]

        preperiodic = set()
        while points != []:
            P = points.pop()
            preimages = self.rational_preimages(P)
            for i in range(len(preimages)):
                if not preimages[i] in preperiodic:
                    points.append(preimages[i])
                    preperiodic.add(preimages[i])
        return list(preperiodic)

    def rational_preperiodic_points(self, **kwds):
        r"""
        Determine the set of rational preperiodic points for
        this dynamical system.

        The map must be defined over `\QQ` and be an endomorphism of
        projective space. If the map is a polynomial endomorphism of
        `\mathbb{P}^1`, i.e. has a totally ramified fixed point, then
        the base ring can be an absolute number field.
        This is done by passing to the Weil restriction.

        The default parameter values are typically good choices for
        `\mathbb{P}^1`. If you are having trouble getting a particular
        map to finish, try first computing the possible periods, then
        try various different values for ``lifting_prime``.

        ALGORITHM:

        - Determines the list of possible periods.

        - Determines the rational periodic points from the possible periods.

        - Determines the rational preperiodic points from the rational
          periodic points by determining rational preimages.

        INPUT:

        kwds:

        - ``prime_bound`` -- (default: ``[1, 20]``) a pair (list or tuple)
          of positive integers that represent the limits of primes to use
          in the reduction step or an integer that represents the upper bound

        - ``lifting_prime`` -- (default: 23) a prime integer; specifies
          modulo which prime to try and perform the lifting

        - ``periods`` -- (optional) a list of positive integers that is
          the list of possible periods

        - ``bad_primes`` -- (optional) a list or tuple of integer primes;
          the primes of bad reduction

        - ``ncpus`` -- (default: all cpus) number of cpus to use in parallel

        OUTPUT: a list of rational points in projective space

        EXAMPLES::

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

        ::

            sage: PS.<x,y> = ProjectiveSpace(1,QQ)
            sage: f = DynamicalSystem_projective([5*x^3 - 53*x*y^2 + 24*y^3, 24*y^3])
            sage: sorted(f.rational_preperiodic_points(prime_bound=10))
            [(-1 : 1), (0 : 1), (1 : 0), (1 : 1), (3 : 1)]

        ::

            sage: PS.<x,y,z> = ProjectiveSpace(2,QQ)
            sage: f = DynamicalSystem_projective([x^2 - 21/16*z^2, y^2-2*z^2, z^2])
            sage: sorted(f.rational_preperiodic_points(prime_bound=[1,8], lifting_prime=7, periods=[2])) # long time
            [(-5/4 : -2 : 1), (-5/4 : -1 : 1), (-5/4 : 0 : 1), (-5/4 : 1 : 1), (-5/4
            : 2 : 1), (-1/4 : -2 : 1), (-1/4 : -1 : 1), (-1/4 : 0 : 1), (-1/4 : 1 :
            1), (-1/4 : 2 : 1), (1/4 : -2 : 1), (1/4 : -1 : 1), (1/4 : 0 : 1), (1/4
            : 1 : 1), (1/4 : 2 : 1), (5/4 : -2 : 1), (5/4 : -1 : 1), (5/4 : 0 : 1),
            (5/4 : 1 : 1), (5/4 : 2 : 1)]

        ::

            sage: K.<w> = QuadraticField(33)
            sage: PS.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2-71/48*y^2, y^2])
            sage: sorted(f.rational_preperiodic_points()) # long time
            [(-1/12*w - 1 : 1),
             (-1/6*w - 1/4 : 1),
             (-1/12*w - 1/2 : 1),
             (-1/6*w + 1/4 : 1),
             (1/12*w - 1 : 1),
             (1/12*w - 1/2 : 1),
             (-1/12*w + 1/2 : 1),
             (-1/12*w + 1 : 1),
             (1/6*w - 1/4 : 1),
             (1/12*w + 1/2 : 1),
             (1 : 0),
             (1/6*w + 1/4 : 1),
             (1/12*w + 1 : 1)]
        """
        PS = self.domain()
        K = PS.base_ring()
        if K not in NumberFields() or not K.is_absolute():
            raise TypeError("base field must be an absolute field")
        d = K.absolute_degree()
        #check that we are not over QQ
        if d > 1:
            if PS.dimension_relative() != 1:
                raise NotImplementedError("rational preperiodic points for number fields only implemented in dimension 1")
            w = K.absolute_generator()
            #we need to dehomogenize for the Weil restriction and will check that point at infty
            #separately. We also check here that we are working with a polynomial. If the map
            #is not a polynomial, the Weil restriction will not be a morphism and we cannot
            #apply this algorithm.
            g = self.dehomogenize(1)
            inf = PS([1,0])
            k = 1
            if isinstance(g[0], FractionFieldElement):
                g = self.dehomogenize(0)
                inf = PS([0,1])
                k = 0
                if isinstance(g[0], FractionFieldElement):
                    raise NotImplementedError("rational preperiodic points for number fields only implemented for polynomials")
            #determine rational preperiodic points
            #infinity is a totally ramified fixed point for a polynomial
            preper = set([inf])
            #compute the weil restriction
            G = g.weil_restriction()
            F = G.homogenize(d)
            #find the QQ rational preperiodic points for the weil restriction
            Fpre = F.rational_preperiodic_points(**kwds)
            for P in Fpre:
                #take the 'good' points in the weil restriction and find the
                #associated number field points.
                if P[d] == 1:
                    pt = [sum([P[i]*w**i for i in range(d)])]
                    pt.insert(k,1)
                    Q = PS(pt)
                    #for each preperiodic point get the entire connected component
                    if not Q in preper:
                        for t in self.connected_rational_component(Q):
                            preper.add(t)
            preper = list(preper)
        else:
            #input error checking done in possible_periods and rational_periodic_points
            badprimes = kwds.pop("bad_primes", None)
            periods = kwds.pop("periods", None)
            primebound = kwds.pop("prime_bound", [1, 20])
            num_cpus = kwds.pop("ncpus", ncpus())
            if badprimes is None:
                badprimes = self.primes_of_bad_reduction()
            if periods is None:
                #determine the set of possible periods
                periods = self.possible_periods(prime_bound=primebound,
                                                bad_primes=badprimes, ncpus=num_cpus)
            if periods == []:
                return([]) #no rational preperiodic points
            else:
                p = kwds.pop("lifting_prime", 23)
                #find the rational preperiodic points
                T = self.rational_periodic_points(prime_bound=primebound, lifting_prime=p,
                                                  periods=periods, bad_primes=badprimes,
                                                  ncpus=num_cpus)
                preper = self.all_rational_preimages(T) #find the preperiodic points
                preper = list(preper)
        return preper

    def rational_preperiodic_graph(self, **kwds):
        r"""
        Determine the directed graph of the rational preperiodic points
        for this dynamical system.

        The map must be defined over `\QQ` and be an endomorphism of
        projective space. If this map is a polynomial endomorphism of
        `\mathbb{P}^1`, i.e. has a totally ramified fixed point, then
        the base ring can be an absolute number field.
        This is done by passing to the Weil restriction.

        ALGORITHM:

        - Determines the list of possible periods.

        - Determines the rational periodic points from the possible periods.

        - Determines the rational preperiodic points from the rational
          periodic points by determining rational preimages.

        INPUT:

        kwds:

        - ``prime_bound`` -- (default: ``[1, 20]``) a pair (list or tuple)
          of positive integers that represent the limits of primes to use
          in the reduction step or an integer that represents the upper bound

        - ``lifting_prime`` -- (default: 23) a prime integer; specifies
          modulo which prime to try and perform the lifting

        - ``periods`` -- (optional) a list of positive integers that is
          the list of possible periods

        - ``bad_primes`` -- (optional) a list or tuple of integer primes;
          the primes of bad reduction

        - ``ncpus`` -- (default: all cpus) number of cpus to use in parallel

        OUTPUT:

        A digraph representing the orbits of the rational preperiodic
        points in projective space.

        EXAMPLES::

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

        ::

            sage: PS.<x,y> = ProjectiveSpace(1,QQ)
            sage: f = DynamicalSystem_projective([-3/2*x^3 +19/6*x*y^2, y^3])
            sage: f.rational_preperiodic_graph(prime_bound=[1,8])
            Looped digraph on 12 vertices

        ::

            sage: PS.<x,y,z> = ProjectiveSpace(2,QQ)
            sage: f = DynamicalSystem_projective([2*x^3 - 50*x*z^2 + 24*z^3,
            ....:                                 5*y^3 - 53*y*z^2 + 24*z^3, 24*z^3])
            sage: f.rational_preperiodic_graph(prime_bound=[1,11], lifting_prime=13) # long time
            Looped digraph on 30 vertices

        ::

            sage: K.<w> = QuadraticField(-3)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2])
            sage: f.rational_preperiodic_graph() # long time
            Looped digraph on 5 vertices
        """
        #input checking done in .rational_preperiodic_points()
        preper = self.rational_preperiodic_points(**kwds)
        g = self._preperiodic_points_to_cyclegraph(preper)
        return g

    def connected_rational_component(self, P, n=0):
        r"""
        Computes the connected component of a rational preperiodic
        point ``P`` by this dynamical system.

        Will work for non-preperiodic points if ``n`` is positive.
        Otherwise this will not terminate.

        INPUT:

        - ``P`` -- a rational preperiodic point of this map

        - ``n`` -- (default: 0) integer; maximum distance from ``P`` to
          branch out; a value of 0 indicates no bound

        OUTPUT:

        A list of points connected to ``P`` up to the specified distance.

        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: f = DynamicalSystem_projective([x^2 - 29/16*y^2, y^2])
            sage: P = PS([w,1])
            sage: f.connected_rational_component(P)
            [(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)]

        ::

            sage: PS.<x,y,z> = ProjectiveSpace(2,QQ)
            sage: f = DynamicalSystem_projective([x^2 - 21/16*z^2, y^2-2*z^2, z^2])
            sage: P = PS([17/16,7/4,1])
            sage: f.connected_rational_component(P,3)
            [(17/16 : 7/4 : 1),
             (-47/256 : 17/16 : 1),
             (-83807/65536 : -223/256 : 1),
             (-17/16 : -7/4 : 1),
             (-17/16 : 7/4 : 1),
             (17/16 : -7/4 : 1),
             (1386468673/4294967296 : -81343/65536 : 1),
             (-47/256 : -17/16 : 1),
             (47/256 : -17/16 : 1),
             (47/256 : 17/16 : 1),
             (-1/2 : -1/2 : 1),
             (-1/2 : 1/2 : 1),
             (1/2 : -1/2 : 1),
             (1/2 : 1/2 : 1)]

        """
        points = [[],[]] # list of points and a list of their corresponding levels
        points[0].append(P)
        points[1].append(0) # P is treated as level 0

        nextpoints = []
        nextpoints.append(P)

        level = 1
        foundall = False # whether done or not
        while not foundall:
            newpoints = []
            for Q in nextpoints:
                # forward image
                newpoints.append(self(Q))
                # preimages
                newpoints.extend(self.rational_preimages(Q))
            del nextpoints[:] # empty list
            # add any points that are not already in the connected component
            for Q in newpoints:
                if (Q not in points[0]):
                    points[0].append(Q)
                    points[1].append(level)
                    nextpoints.append(Q)
            # done if max level was achieved or if there were no more points to add
            if ((level + 1 > n and n != 0) or len(nextpoints) == 0):
                foundall = True
            level = level + 1

        return points[0]

    def conjugating_set(self, other):
        r"""
        Return the set of elements in PGL that conjugates one
        dynamical system to the other.

        Given two nonconstant rational functions of equal degree
        determine to see if there is an element of PGL that
        conjugates one rational function to another. It does this
        by taking the fixed points of one map and mapping
        them to all unique permutations of the fixed points of
        the other map. If there are not enough fixed points the
        function compares the mapping between rational preimages of
        fixed points and the rational preimages of the preimages of
        fixed points until there are enough points; such that there
        are `n+2` points with all `n+1` subsets linearly independent.

        ALGORITHM:

        Implementing invariant set algorithim from the paper [FMV2014]_.
        Given that the set of  `n` th preimages of fixed points is
        invariant under conjugation find all elements of PGL that
        take one set to another.

        INPUT:

        - ``other`` -- a nonconstant rational function of same degree
          as ``self``

        OUTPUT:

        Set of conjugating `n+1` by `n+1` matrices.

        AUTHORS:

        - Original algorithm written by Xander Faber, Michelle Manes,
          Bianca Viray [FMV2014]_.

        - Implimented by Rebecca Lauren Miller, as part of GSOC 2016.

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 - 2*y^2, y^2])
            sage: m = matrix(QQbar, 2, 2, [-1, 3, 2, 1])
            sage: g = f.conjugate(m)
            sage: f.conjugating_set(g)
            [
            [-1  3]
            [ 2  1]
            ]

        ::

            sage: K.<w> = QuadraticField(-1)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^2 + y^2, x*y])
            sage: m = matrix(K, 2, 2, [1, 1, 2, 1])
            sage: g = f.conjugate(m)
            sage: f.conjugating_set(g) # long time
            [
            [1 1]  [-1 -1]
            [2 1], [ 2  1]
            ]

        ::

            sage: K.<i> = QuadraticField(-1)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: D8 = DynamicalSystem_projective([y^3, x^3])
            sage: D8.conjugating_set(D8) # long time
            [
            [1 0]  [0 1]  [ 0 -i]  [i 0]  [ 0 -1]  [-1  0]  [-i  0]  [0 i]
            [0 1], [1 0], [ 1  0], [0 1], [ 1  0], [ 0  1], [ 0  1], [1 0]
            ]

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: D8 = DynamicalSystem_projective([y^2, x^2])
            sage: D8.conjugating_set(D8)
            Traceback (most recent call last):
            ...
            ValueError: not enough rational preimages

        ::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: f = DynamicalSystem_projective([x^2 + x*z, y^2, z^2])
            sage: f.conjugating_set(f) # long time
            [
            [1 0 0]
            [0 1 0]
            [0 0 1]
            ]
        """
        f = copy(self)
        g = copy(other)
        try:
            f.normalize_coordinates()
            g.normalize_coordinates()
        except (ValueError):
            pass
        if f.degree() != g.degree():# checks that maps are of equal degree
            return []
        n = f.domain().dimension_relative()
        L = Set(f.periodic_points(1))
        K = Set(g.periodic_points(1))
        if len(L) != len(K):  # checks maps have the same number of fixed points
            return []
        d = len(L)
        r = f.domain().base_ring()
        more = True
        if d >= n+2: # need at least n+2 points
            for i in Subsets(L, n+2):
                # make sure all n+1 subsets are linearly independent
                Ml = matrix(r, [list(s) for s in i])
                if not any([j == 0 for j in Ml.minors(n+1)]):
                    Tf = list(i)
                    more= False
                    break
        while more:
            #  finds preimages of fixed points
            Tl = [Q for i in L for Q in f.rational_preimages(i)]
            Tk = [Q for i in K for Q in g.rational_preimages(i)]
            if len(Tl) != len(Tk):
                return []
            L = L.union(Set(Tl))
            K = K.union(Set(Tk))
            if d == len(L): # if no new preimages then not enough points
                raise ValueError("not enough rational preimages")
            d = len(L)
            if d >= n + 2: # makes sure all n+1 subsets are linearly independent
                for i in Subsets(L, n+2):
                    Ml = matrix(r, [list(s) for s in i])
                    if not any([j == 0 for j in Ml.minors(n+1)]):
                        more = False
                        Tf = list(i)
                        break
        Conj = []
        for i in Arrangements(K,(n+2)):
            # try all possible conjugations between invariant sets
            try: # need all n+1 subsets linearly independenet
                s = f.domain().point_transformation_matrix(i,Tf)# finds elements of PGL that maps one map to another
                if self.conjugate(s) == other:
                    Conj.append(s)
            except (ValueError):
                pass
        return Conj

    def is_conjugate(self, other):
        r"""
        Return whether or not two dynamcial systems are conjugate.

        ALGORITHM:

        Implementing invariant set algorithim from the paper [FMV2014]_.
        Given that the set of `n` th preimages is invariant under
        conjugation this function finds whether two maps are conjugate.

        INPUT:

        - ``other`` -- a nonconstant rational function of same degree
          as ``self``

        OUTPUT: boolean

        AUTHORS:

        - Original algorithm written by Xander Faber, Michelle Manes,
          Bianca Viray [FMV2014]_.

        - Implimented by Rebecca Lauren Miller as part of GSOC 2016.

        EXAMPLES::

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

        ::

            sage: set_verbose(None)
            sage: P.<x,y> = ProjectiveSpace(QQbar,1)
            sage: f = DynamicalSystem_projective([x^2 + x*y,y^2])
            sage: m = matrix(QQbar, 2, 2, [1, 1, 2, 1])
            sage: g = f.conjugate(m)
            sage: f.is_conjugate(g) # long time
            True

        ::

            sage: P.<x,y> = ProjectiveSpace(GF(5),1)
            sage: f = DynamicalSystem_projective([x^3 + x*y^2,y^3])
            sage: m = matrix(GF(5), 2, 2, [1, 3, 2, 9])
            sage: g = f.conjugate(m)
            sage: f.is_conjugate(g)
            True

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + x*y,y^2])
            sage: g = DynamicalSystem_projective([x^3 + x^2*y, y^3])
            sage: f.is_conjugate(g)
            False

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([x^2 + x*y, y^2])
            sage: g = DynamicalSystem_projective([x^2 - 2*y^2, y^2])
            sage: f.is_conjugate(g)
            False
        """
        f = copy(self)
        g = copy(other)
        try:
            f.normalize_coordinates()
            g.normalize_coordinates()
        except (ValueError):
            pass
        if f.degree() != g.degree(): # checks that maps are of equal degree
            return False
        n = f.domain().dimension_relative()
        L = Set(f.periodic_points(1))
        K = Set(g.periodic_points(1))
        if len(L) != len(K): # checks maps have the same number of fixed points
            return False
        d = len(L)
        r = f.domain().base_ring()
        more = True
        if d >= n+2: # need at least n+2 points
            for i in Subsets(L, n+2): # makes sure all n+1 subsets are linearly independent
                Ml = matrix(r, [list(s) for s in i])
                if not any([j == 0 for j in Ml.minors(n+1)]):
                    Tf = list(i)
                    more = False
                    break
        while more:
            # finds preimages of fixed points
            Tl = [Q for i in L for Q in f.rational_preimages(i)]
            Tk = [Q for i in K for Q in g.rational_preimages(i)]
            if len(Tl) != len(Tk):
                return False
            L = L.union(Set(Tl))
            K = K.union(Set(Tk))
            if d == len(L):# if no new preimages then not enough points
                raise ValueError("not enough rational preimages")
            d = len(L)
            if d >= n + 2:
                # make sure all n+1 subsets are linearly independent
                for i in Subsets(L, n+2): # checks at least n+1 are linearly independent
                    Ml = matrix(r, [list(s) for s in i])
                    if not any([j == 0 for j in Ml.minors(n+1)]):
                        more = False
                        Tf = list(i)
                        break
        for i in Arrangements(K, n+2):
            # try all possible conjugations between invariant sets
            try: # need all n+1 subsets linearly independenet
                s = f.domain().point_transformation_matrix(i,Tf) # finds elements of PGL that maps one map to another
                if self.conjugate(s) == other:
                    return True
            except (ValueError):
                pass
        return False

    def is_polynomial(self):
        r"""
        Check to see if the dynamcial system has a totally ramified
        fixed point.

        The function must be defined over an absolute number field or a
        finite field.

        OUTPUT: boolean

        EXAMPLES::

            sage: R.<x> = QQ[]
            sage: K.<w> = QuadraticField(7)
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: f = DynamicalSystem_projective([x**2 + 2*x*y - 5*y**2, 2*x*y])
            sage: f.is_polynomial()
            False

        ::

            sage: R.<x> = QQ[]
            sage: K.<w> = QuadraticField(7)
            sage: P.<x,y> = ProjectiveSpace(K, 1)
            sage: f = DynamicalSystem_projective([x**2 - 7*x*y, 2*y**2])
            sage: m = matrix(K, 2, 2, [w, 1, 0, 1])
            sage: f = f.conjugate(m)
            sage: f.is_polynomial()
            True

        ::

            sage: K.<w> = QuadraticField(4/27)
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x**3 + w*y^3,x*y**2])
            sage: f.is_polynomial()
            False

        ::

            sage: K = GF(3**2, prefix='w')
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x**2 + K.gen()*y**2, x*y])
            sage: f.is_polynomial()
            False

        ::

            sage: PS.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([6*x^2+12*x*y+7*y^2, 12*x*y + 42*y^2])
            sage: f.is_polynomial()
            False
        """
        if self.codomain().dimension_relative() != 1:
            raise NotImplementedError("space must have dimension equal to 1")
        K = self.base_ring()
        if not K in FiniteFields() and (not K in NumberFields() or not K.is_absolute()):
            raise NotImplementedError("must be over an absolute number field or finite field")
        if K in FiniteFields():
            q = K.characteristic()
            deg = K.degree()
            var = K.variable_name()
        g = self
        #get polynomial defining fixed points
        G = self.dehomogenize(1).dynatomic_polynomial(1)
        # see if infty = (1,0) is fixed
        if G.degree() <= g.degree():
            #check if infty is totally ramified
            if len((g[1]).factor()) == 1:
                return True
        #otherwise we need to create the tower of extensions
        #which contain the fixed points. We do
        #this successively so we can exit early if
        #we find one and not go all the way to the splitting field
        i = 0 #field index
        if G.degree() != 0:
            G = G.polynomial(G.variable(0))
        while G.degree() != 0:
            Y = G.factor()
            R = G.parent()
            u = G
            for p,e in Y:
                if p.degree() == 1:
                    if len((g[0]*p[1] + g[1]*p[0]).factor()) == 1:
                        return True
                    G = R(G/p) # we already checked this root
                else:
                    u = p #need to extend to get these roots
            if G.degree() != 0:
                #create the next extension
                if K == QQ:
                    from sage.rings.number_field.number_field import NumberField
                    L = NumberField(u, 't'+str(i))
                    i += 1
                    phi = K.embeddings(L)[0]
                    K = L
                elif K in FiniteFields():
                    deg = deg*G.degree()
                    K = GF(q**(deg), prefix=var)
                else:
                    L = K.extension(u, 't'+str(i))
                    i += 1
                    phi1 = K.embeddings(L)[0]
                    K = L
                    L = K.absolute_field('t'+str(i))
                    i += 1
                    phi = K.embeddings(L)[0]*phi1
                    K = L
                if K in FiniteFields():
                    G = G.change_ring(K)
                    g = g.change_ring(K)
                else:
                    G = G.change_ring(phi)
                    g = g.change_ring(phi)
        return False

    def normal_form(self, return_conjugation=False):
        r"""
        Return a normal form in the moduli space of dynamical systems.

        Currently implemented only for polynomials. The totally ramified
        fixed point is moved to infinity and the map is conjugated to the form
        `x^n + a_{n-2} x^{n-2} + \cdots + a_{0}`. Note that for finite fields
        we can only remove the `(n-1)`-st term when the characteristic
        does not divide `n`.

        INPUT:

        - ``return_conjugation`` -- (default: ``False``) boolean; if ``True``,
          then return the conjugation element of PGL along with the embedding
          into the new field

        OUTPUT:

        - :class:`SchemeMorphism_polynomial`

        - (optional) an element of PGL as a matrix

        - (optional) the field embedding

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ, 1)
            sage: f = DynamicalSystem_projective([x^2 + 2*x*y - 5*x^2, 2*x*y])
            sage: f.normal_form()
            Traceback (most recent call last):
            ...
            NotImplementedError: map is not a polynomial

        ::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: f = DynamicalSystem_projective([13*x^2 + 4*x*y + 3*y^2, 5*y^2])
            sage: f.normal_form()
            Dynamical System of Projective Space of dimension 1 over Rational Field
              Defn: Defined on coordinates by sending (x : y) to
                    (5*x^2 + 9*y^2 : 5*y^2)

        ::

            sage: K = GF(3^3, prefix = 'w')
            sage: P.<x,y> = ProjectiveSpace(K,1)
            sage: f = DynamicalSystem_projective([x^3 + 2*x^2*y + 2*x*y^2 + K.gen()*y^3, y^3])
            sage: f.normal_form()
            Dynamical System of Projective Space of dimension 1 over Finite Field in w3 of size 3^3
                  Defn: Defined on coordinates by sending (x : y) to
                        (x^3 + x^2*y + x*y^2 + (-w3)*y^3 : y^3)
        """
        #defines the field of fixed points
        if self.codomain().dimension_relative() != 1:
            raise NotImplementedError("space must have dimension equal to 1")
        K = self.base_ring()
        if not K in FiniteFields() and (not K in NumberFields() or not K.is_absolute()):
            raise NotImplementedError("must be over an absolute number field or finite field")
        if K in FiniteFields():
            q = K.characteristic()
            deg = K.degree()
            var = K.variable_name()
        else:
            psi = K.hom([K.gen()]) #identity hom for return_embedding
        g = self
        G = self.dehomogenize(1).dynatomic_polynomial(1)
        done = False
        bad = True
        #check infty = (1,0) is fixed
        if G.degree() <= g.degree():
            #check infty totally ramified
            if len((g[1]).factor()) == 1:
                T = self.domain()(1,0)
                bad = False
                done = True
                m = matrix(K, 2, 2, [1,0,0,1])
        #otherwise we need to create the tower of extensions
        #which contain the fixed points. We do
        #this successively so we can early exit if
        #we find one and not go all the way to the splitting field
        i = 0
        if G.degree() != 0:
            G = G.polynomial(G.variable(0))
        else:
            #no other fixed points
            raise NotImplementedError("map is not a polynomial")
        #check other fixed points
        while not done:
            Y = G.factor()
            R = G.parent()
            done = True
            for p,e in Y:
                if p.degree() == 1:
                    if len((g[0]*p[1] + g[1]*p[0]).factor()) == 1:
                        T = self.domain()(-p[0], p[1])
                        bad = False
                        done = True
                        break # bc only 1 totally ramified fixed pt
                    G = R(G/p)
                else:
                    done = False
                    u = p
            if not done:
                #extend
                if K == QQ:
                    from sage.rings.number_field.number_field import NumberField
                    L = NumberField(u, 't'+str(i))
                    i += 1
                    phi = K.embeddings(L)[0]
                    psi = phi * psi
                    K = L
                elif K in FiniteFields():
                    deg = deg * G.degree()
                    K = GF(q**(deg), prefix=var)
                else:
                    L = K.extension(u, 't'+str(i))
                    i += 1
                    phi1 = K.embeddings(L)[0]
                    K = L
                    L = K.absolute_field('t'+str(i))
                    i += 1
                    phi = K.embeddings(L)[0]*phi1
                    psi = phi * psi
                    K = L
                #switch to the new field
                if K in FiniteFields():
                    G = G.change_ring(K)
                    g = g.change_ring(K)
                else:
                    G = G.change_ring(phi)
                    g = g.change_ring(phi)
        if bad:
            raise NotImplementedError("map is not a polynomial")
        #conjugate to normal form
        Q = T.codomain()
        #moved totally ramified fixed point to infty
        target = [T, Q(T[0]+1, 1), Q(T[0]+2, 1)]
        source = [Q(1, 0), Q(0, 1), Q(1, 1)]
        m = Q.point_transformation_matrix(source, target)
        N = g.base_ring()
        d = g.degree()
        gc = g.conjugate(m)
        #make monic
        R = PolynomialRing(N, 'z')
        v = N(gc[1].coefficient([0,d])/gc[0].coefficient([d,0]))
        #need a (d-1)-st root to make monic
        u = R.gen(0)**(d-1) - v
        if d != 2 and u.is_irreducible():
            #we need to extend again
            if N in FiniteFields():
                deg = deg*(d-1)
                M = GF(q**(deg), prefix=var)
            else:
                L = N.extension(u,'t'+str(i))
                i += 1
                phi1 = N.embeddings(L)[0]
                M = L.absolute_field('t'+str(i))
                phi = L.embeddings(M)[0]*phi1
                psi = phi*psi
            if M in FiniteFields():
                gc = gc.change_ring(M)
            else:
                gc = gc.change_ring(phi)
            m = matrix(M, 2, 2, [phi(s) for t in list(m) for s in t])
            rv = phi(v).nth_root(d-1)
        else: #root is already in the field
            M = N
            rv = v.nth_root(d-1)
        mc = matrix(M, 2, 2, [rv,0,0,1])
        gcc = gc.conjugate(mc)
        if not (M in FiniteFields() and q.divides(d)):
            #remove 2nd order term
            mc2 = matrix(M, 2, 2, [1, M((-gcc[0].coefficient([d-1, 1])
                / (d*gcc[1].coefficient([0, d]))).constant_coefficient()), 0, 1])
        else:
            mc2 = mc.parent().one()
        gccc = gcc.conjugate(mc2)
        if return_conjugation:
            if M in FiniteFields():
                return gccc, m * mc * mc2
            else:
                return gccc, m * mc * mc2, psi
        return gccc

class DynamicalSystem_projective_finite_field(DynamicalSystem_projective_field,
                                              SchemeMorphism_polynomial_projective_space_finite_field):
    def orbit_structure(self, P):
        r"""
        Return the pair ``[m,n]``, where ``m`` is the preperiod and ``n``
        is the period of the point ``P`` by this dynamical system.

        Every point is preperiodic over a finite field so every point
        will be preperiodic.

        INPUT:

        - ``P`` -- a point in the domain of this map

        OUTPUT: a list ``[m,n]`` of integers

        EXAMPLES::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(GF(7),2)
            sage: X = P.subscheme(x^2-y^2)
            sage: f = DynamicalSystem_projective([x^2, y^2, z^2], domain=X)
            sage: f.orbit_structure(X(1,1,2))
            [0, 2]

        ::

            sage: P.<x,y> = ProjectiveSpace(GF(13),1)
            sage: f = DynamicalSystem_projective([x^2 - y^2, y^2], domain=P)
            sage: f.orbit_structure(P(3,4))
            [2, 3]

        ::

            sage: R.<t> = GF(13^3)
            sage: P.<x,y> = ProjectiveSpace(R,1)
            sage: f = DynamicalSystem_projective([x^2 - y^2, y^2], domain=P)
            sage: f.orbit_structure(P(t, 4))
            [11, 6]
        """
        orbit = []
        index = 1
        Q = copy(P)
        Q.normalize_coordinates()
        F = copy(self)
        F.normalize_coordinates()
        while not Q in orbit:
            orbit.append(Q)
            Q = F(Q)
            Q.normalize_coordinates()
            index += 1
        I = orbit.index(Q)
        return([I, index-I-1])

    def cyclegraph(self):
        r"""
        Return the digraph of all orbits of this dyanmical system.

        Over a finite field this is a finite graph. For subscheme domains, only points
        on the subscheme whose image are also on the subscheme are in the digraph.

        OUTPUT: a digraph

        EXAMPLES::

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

        ::

            sage: P.<x,y,z> = ProjectiveSpace(GF(3^2,'t'),2)
            sage: f = DynamicalSystem_projective([x^2+y^2, y^2, z^2+y*z])
            sage: f.cyclegraph()
            Looped digraph on 91 vertices

        ::

            sage: P.<x,y,z> = ProjectiveSpace(GF(7),2)
            sage: X = P.subscheme(x^2-y^2)
            sage: f = DynamicalSystem_projective([x^2, y^2, z^2], domain=X)
            sage: f.cyclegraph()
            Looped digraph on 15 vertices

        ::

            sage: P.<x,y,z> = ProjectiveSpace(GF(3),2)
            sage: f = DynamicalSystem_projective([x*z-y^2, x^2-y^2, y^2-z^2])
            sage: f.cyclegraph()
            Looped digraph on 13 vertices

        ::

            sage: P.<x,y,z> = ProjectiveSpace(GF(3),2)
            sage: X = P.subscheme([x-y])
            sage: f = DynamicalSystem_projective([x^2-y^2, x^2-y^2, y^2-z^2], domain=X)
            sage: f.cyclegraph()
            Looped digraph on 4 vertices
        """
        V = []
        E = []
        if is_ProjectiveSpace(self.domain()):
            for P in self.domain():
                V.append(P)
                try:
                    Q = self(P)
                    Q.normalize_coordinates()
                    E.append([Q])
                except ValueError: #indeterminacy
                    E.append([])
        else:
            X = self.domain()
            for P in X.ambient_space():
                try:
                    XP = X.point(P)
                    V.append(XP)
                    try:
                        Q = self(XP)
                        Q.normalize_coordinates()
                        E.append([Q])
                    except ValueError: #indeterminacy
                        E.append([])
                except TypeError:  # not a point on the scheme
                    pass
        from sage.graphs.digraph import DiGraph
        g = DiGraph(dict(zip(V, E)), loops=True)
        return g

    def possible_periods(self, return_points=False):
        r"""
        Return the list of possible minimal periods of a periodic point
        over `\QQ` and (optionally) a point in each cycle.

        ALGORITHM:

        See [Hutz2009]_.

        INPUT:

        - ``return_points`` -- (default: ``False``) boolean; if ``True``,
          then return the points as well as the possible periods

        OUTPUT:

        A list of positive integers, or a list of pairs of projective
        points and periods if ``return_points`` is ``True``.

        EXAMPLES::

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

        ::

            sage: P.<x,y> = ProjectiveSpace(GF(13),1)
            sage: f = DynamicalSystem_projective([x^2-y^2, y^2])
            sage: sorted(f.possible_periods(True))
            [[(0 : 1), 2], [(1 : 0), 1], [(3 : 1), 3], [(3 : 1), 36]]

        ::

            sage: PS.<x,y,z> = ProjectiveSpace(2,GF(7))
            sage: f = DynamicalSystem_projective([-360*x^3 + 760*x*z^2,
            ....:                                 y^3 - 604*y*z^2 + 240*z^3, 240*z^3])
            sage: f.possible_periods()
            [1, 2, 4, 6, 12, 14, 28, 42, 84]

        .. TODO::

            - do not return duplicate points

            - improve hash to reduce memory of point-table
        """
        return _fast_possible_periods(self, return_points)

    def automorphism_group(self, absolute=False, iso_type=False, return_functions=False):
        r"""
        Return the subgroup of `PGL2` that is the automorphism group of this
        dynamical system.

        Only for dimension 1. The automorphism group is the set of `PGL2`
        elements that fixed the map under conjugation. See [FMV2014]_
        for the algorithm.

        INPUT:

        - ``absolute``-- (default: ``False``) boolean; if ``True``, then
          return the absolute automorphism group and a field of definition

        - ``iso_type`` -- (default: ``False``) boolean; if ``True``, then
          return the isomorphism type of the automorphism group

        - ``return_functions`` -- (default: ``False``) boolean; ``True``
          returns elements as linear fractional transformations and
          ``False`` returns elements as `PGL2` matrices

        OUTPUT: a list of elements of the automorphism group

        AUTHORS:

        - Original algorithm written by Xander Faber, Michelle Manes,
          Bianca Viray

        - Modified by Joao Alberto de Faria, Ben Hutz, Bianca Thompson

        EXAMPLES::

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

        ::

            sage: R.<x,y> = ProjectiveSpace(GF(3^2,'t'),1)
            sage: f = DynamicalSystem_projective([x^3,y^3])
            sage: f.automorphism_group(return_functions=True, iso_type=True) # long time
            ([x, x/(x + 1), x/(2*x + 1), 2/(x + 2), (2*x + 1)/(2*x), (2*x + 2)/x,
            1/(2*x + 2), x + 1, x + 2, x/(x + 2), 2*x/(x + 1), 2*x, 1/x, 2*x + 1,
            2*x + 2, ((t + 2)*x + t + 2)/((2*t + 1)*x + t + 2), (t*x + 2*t)/(t*x +
            t), 2/x, (x + 1)/(x + 2), (2*t*x + t)/(t*x), (2*t + 1)/((2*t + 1)*x +
            2*t + 1), ((2*t + 1)*x + 2*t + 1)/((2*t + 1)*x), t/(t*x + 2*t), (2*x +
            1)/(x + 1)], 'PGL(2,3)')

        ::

            sage: R.<x,y> = ProjectiveSpace(GF(2^5,'t'),1)
            sage: f = DynamicalSystem_projective([x^5,y^5])
            sage: f.automorphism_group(return_functions=True, iso_type=True)
            ([x, 1/x], 'Cyclic of order 2')

        ::

            sage: R.<x,y> = ProjectiveSpace(GF(3^4,'t'),1)
            sage: f = DynamicalSystem_projective([x^2+25*x*y+y^2, x*y+3*y^2])
            sage: f.automorphism_group(absolute=True)
            [Univariate Polynomial Ring in w over Finite Field in b of size 3^4,
             [
             [1 0]
             [0 1]
             ]]
        """
        if self.domain().dimension_relative() != 1:
            raise NotImplementedError("must be dimension 1")
        else:
            f = self.dehomogenize(1)
            z = f[0].parent().gen()
        if f[0].denominator() != 1:
            F = f[0].numerator().polynomial(z) / f[0].denominator().polynomial(z)
        else:
            F = f[0].numerator().polynomial(z)
        from .endPN_automorphism_group import automorphism_group_FF
        return(automorphism_group_FF(F, absolute, iso_type, return_functions))

