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affine_morphism.py
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affine_morphism.py
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r"""
Morphisms on affine varieties
A morphism of schemes determined by rational functions that define \
what the morphism does on points in the ambient affine space.
AUTHORS:
- David Kohel, William Stein
- Volker Braun (2011-08-08): Renamed classes, more documentation, misc
cleanups.
- Ben Hutz (2013-03) iteration functionality and new directory structure
for affine/projective
"""
#*****************************************************************************
# 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 sage.calculus.functions import jacobian
from sage.categories.fields import Fields
from sage.categories.homset import Hom
from sage.matrix.constructor import matrix, identity_matrix
from sage.misc.cachefunc import cached_method
from sage.misc.all import prod
from sage.rings.all import Integer
from sage.arith.all import lcm, gcd
from sage.rings.complex_field import ComplexField
from sage.rings.finite_rings.finite_field_constructor import GF, is_PrimeFiniteField
from sage.rings.fraction_field import FractionField
from sage.rings.fraction_field_element import FractionFieldElement
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.real_mpfr import RealField
from sage.schemes.generic.morphism import SchemeMorphism_polynomial
from sage.misc.lazy_attribute import lazy_attribute
from sage.ext.fast_callable import fast_callable
import sys
class SchemeMorphism_polynomial_affine_space(SchemeMorphism_polynomial):
"""
A morphism of schemes determined by rational functions.
EXAMPLES::
sage: RA.<x,y> = QQ[]
sage: A2 = AffineSpace(RA)
sage: RP.<u,v,w> = QQ[]
sage: P2 = ProjectiveSpace(RP)
sage: H = A2.Hom(P2)
sage: f = H([x, y, 1])
sage: f
Scheme morphism:
From: Affine Space of dimension 2 over Rational Field
To: Projective Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x, y) to
(x : y : 1)
"""
def __init__(self, parent, polys, check=True):
r"""
The Python constructor.
See :class:`SchemeMorphism_polynomial` for details.
INPUT:
- ``parent`` -- Hom.
- ``polys`` -- list or tuple of polynomial or rational functions.
- ``check`` -- Boolean.
OUTPUT:
- :class:`SchemeMorphism_polynomial_affine_space`.
EXAMPLES::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: H = Hom(A, A)
sage: H([3/5*x^2, y^2/(2*x^2)])
Traceback (most recent call last):
...
TypeError: polys (=[3/5*x^2, y^2/(2*x^2)]) must be rational functions in
Multivariate Polynomial Ring in x, y over Integer Ring
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: H = Hom(A, A)
sage: H([3*x^2/(5*y), y^2/(2*x^2)])
Scheme endomorphism of Affine Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x, y) to
(3*x^2/(5*y), y^2/(2*x^2))
::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: H = Hom(A, A)
sage: H([3/2*x^2, y^2])
Scheme endomorphism of Affine Space of dimension 2 over Rational Field
Defn: Defined on coordinates by sending (x, y) to
(3/2*x^2, y^2)
::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: X = A.subscheme([x-y^2])
sage: H = Hom(X, X)
sage: H([9/4*x^2, 3/2*y])
Scheme endomorphism of Closed subscheme of Affine Space of dimension 2
over Rational Field defined by:
-y^2 + x
Defn: Defined on coordinates by sending (x, y) to
(9/4*x^2, 3/2*y)
sage: P.<x,y,z> = ProjectiveSpace(ZZ, 2)
sage: H = Hom(P, P)
sage: f = H([5*x^3 + 3*x*y^2-y^3, 3*z^3 + y*x^2, x^3-z^3])
sage: f.dehomogenize(2)
Scheme endomorphism of Affine Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x0, x1) to
((5*x0^3 + 3*x0*x1^2 - x1^3)/(x0^3 - 1), (x0^2*x1 + 3)/(x0^3 - 1))
If you pass in quotient ring elements, they are reduced::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([x-y])
sage: H = Hom(X,X)
sage: u,v,w = X.coordinate_ring().gens()
sage: H([u, v, u+v])
Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
over Rational Field defined by:
x - y
Defn: Defined on coordinates by sending (x, y, z) to
(y, y, 2*y)
You must use the ambient space variables to create rational functions::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([x^2-y^2])
sage: H = Hom(X,X)
sage: u,v,w = X.coordinate_ring().gens()
sage: H([u, v, (u+1)/v])
Traceback (most recent call last):
...
ArithmeticError: Division failed. The numerator is not a multiple of the denominator.
sage: H([x, y, (x+1)/y])
Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
over Rational Field defined by:
x^2 - y^2
Defn: Defined on coordinates by sending (x, y, z) to
(x, y, (x + 1)/y)
::
sage: R.<t> = PolynomialRing(QQ)
sage: A.<x,y,z> = AffineSpace(R, 3)
sage: X = A.subscheme(x^2-y^2)
sage: H = End(X)
sage: H([x^2/(t*y), t*y^2, x*z])
Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
over Univariate Polynomial Ring in t over Rational Field defined by:
x^2 - y^2
Defn: Defined on coordinates by sending (x, y, z) to
(x^2/(t*y), t*y^2, x*z)
"""
if check:
if not isinstance(polys, (list, tuple)):
raise TypeError("polys (=%s) must be a list or tuple"%polys)
source_ring = parent.domain().ambient_space().coordinate_ring()
target = parent.codomain().ambient_space()
if len(polys) != target.ngens():
raise ValueError("there must be %s polynomials"%target.ngens())
try:
polys = [source_ring(poly) for poly in polys]
except TypeError: #maybe given quotient ring elements
try:
polys = [source_ring(poly.lift()) for poly in polys]
except (TypeError, AttributeError):
#must be a rational function since we cannot have
#rational functions for quotient rings
try:
if not all(p.base_ring()==source_ring.base_ring() for p in polys):
raise TypeError("polys (=%s) must be rational functions in %s"%(polys, source_ring))
polys = [source_ring(poly.numerator())/source_ring(poly.denominator()) for poly in polys]
except TypeError: #can't seem to coerce
raise TypeError("polys (=%s) must be rational functions in %s"%(polys, source_ring))
self._is_prime_finite_field = is_PrimeFiniteField(polys[0].base_ring()) # Needed for _fast_eval and _fastpolys
SchemeMorphism_polynomial.__init__(self, parent, polys, False)
def __call__(self, x, check=True):
"""
Evaluate affine morphism at point described by ``x``.
EXAMPLES::
sage: P.<x,y,z> = AffineSpace(QQ, 3)
sage: H = Hom(P, P)
sage: f = H([x^2+y^2, y^2, z^2 + y*z])
sage: f(P([1, 1, 1]))
(2, 1, 2)
"""
from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
if check:
if not isinstance(x, SchemeMorphism_point_affine):
try:
x = self.domain()(x)
except (TypeError, NotImplementedError):
raise TypeError("%s fails to convert into the map's domain %s, but a `pushforward` method is not properly implemented"%(x, self.domain()))
elif self.domain() != x.codomain():
raise TypeError("%s fails to convert into the map's domain %s,but a `pushforward` method is not properly implemented"%(x, self.domain()))
# Passes the array of args to _fast_eval
P = self._fast_eval(x._coords)
return self.codomain().point(P, check)
def __eq__(self, right):
"""
Tests the equality of two affine maps.
INPUT:
- ``right`` - a map on affine space.
OUTPUT:
- Boolean - True if the two affine maps define the same map.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: A2.<u,v> = AffineSpace(QQ, 2)
sage: H = End(A)
sage: H2 = End(A2)
sage: f = H([x^2 - 2*x*y, y/(x+1)])
sage: g = H2([u^3 - v, v^2])
sage: f == g
False
::
sage: A.<x,y,z> = AffineSpace(CC, 3)
sage: H = End(A)
sage: f = H([x^2 - CC.0*x*y + z*x, 1/z^2 - y^2, 5*x])
sage: f == f
True
"""
if not isinstance(right, SchemeMorphism_polynomial):
return False
if self.parent() != right.parent():
return False
return all([self[i] == right[i] for i in range(len(self._polys))])
def __ne__(self, right):
"""
Tests the inequality of two affine maps.
INPUT:
- ``right`` - a map on affine space.
OUTPUT:
- Boolean - True if the two affine maps define the same map.
EXAMPLES::
sage: A.<x,y> = AffineSpace(RR, 2)
sage: H = End(A)
sage: f = H([x^2 - y, y^2])
sage: g = H([x^3-x*y, x*y^2])
sage: f != g
True
sage: f != f
False
"""
if not isinstance(right, SchemeMorphism_polynomial):
return True
if self.parent() != right.parent():
return True
if all([self[i] == right[i] for i in range(len(self._polys))]):
return False
return True
@lazy_attribute
def _fastpolys(self):
"""
Lazy attribute for fast_callable polynomials for affine morphsims.
EXAMPLES::
sage: P.<x,y> = AffineSpace(QQ, 2)
sage: H = Hom(P, P)
sage: f = H([x^2+y^2, y^2/(1+x)])
sage: [t.op_list() for g in f._fastpolys for t in g]
[[('load_const', 0), ('load_const', 1), ('load_arg', 1), ('ipow', 2),
'mul', 'add', ('load_const', 1), ('load_arg', 0), ('ipow', 2), 'mul',
'add', 'return'], [('load_const', 0), ('load_const', 1), ('load_arg',
1), ('ipow', 2), 'mul', 'add', 'return'], [('load_const', 0),
('load_const', 1), 'add', 'return'], [('load_const', 0), ('load_const',
1), ('load_arg', 0), ('ipow', 1), 'mul', 'add', ('load_const', 1),
'add', 'return']]
"""
polys = self._polys
R = self.domain().ambient_space().coordinate_ring()
# fastpolys[0] corresponds to the numerator polys, fastpolys[1] corresponds to denominator polys
fastpolys = [[], []]
for poly in polys:
# Determine if truly polynomials. Store the numerator and denominator as separate polynomials
# and repeat the normal process for both.
try:
poly_numerator = R(poly)
poly_denominator = R.one()
except TypeError:
poly_numerator = R(poly.numerator())
poly_denominator = R(poly.denominator())
# These tests are in place because the float and integer domain evaluate
# faster than using the base_ring
if self._is_prime_finite_field:
prime = polys[0].base_ring().characteristic()
degree = max(poly_numerator.degree(), poly_denominator.degree())
height = max([abs(c.lift()) for c in poly_numerator.coefficients()]\
+ [abs(c.lift()) for c in poly_denominator.coefficients()])
num_terms = max(len(poly_numerator.coefficients()), len(poly_denominator.coefficients()))
largest_value = num_terms * height * (prime - 1) ** degree
# If the calculations will not overflow the float data type use domain float
# Else use domain integer
if largest_value < (2 ** sys.float_info.mant_dig):
fastpolys[0].append(fast_callable(poly_numerator, domain=float))
fastpolys[1].append(fast_callable(poly_denominator, domain=float))
else:
fastpolys[0].append(fast_callable(poly_numerator, domain=ZZ))
fastpolys[1].append(fast_callable(poly_denominator, domain=ZZ))
else:
fastpolys[0].append(fast_callable(poly_numerator, domain=poly.base_ring()))
fastpolys[1].append(fast_callable(poly_denominator, domain=poly.base_ring()))
return fastpolys
def _fast_eval(self, x):
"""
Evaluate affine morphism at point described by ``x``.
EXAMPLES::
sage: P.<x,y,z> = AffineSpace(QQ, 3)
sage: H = Hom(P, P)
sage: f = H([x^2+y^2, y^2, z^2 + y*z])
sage: f._fast_eval([1, 1, 1])
[2, 1, 2]
::
sage: P.<x,y,z> = AffineSpace(QQ, 3)
sage: H = Hom(P, P)
sage: f = H([x^2/y, y/x, (y^2+z)/(x*y)])
sage: f._fast_eval([2, 3, 1])
[4/3, 3/2, 5/3]
"""
R = self.domain().ambient_space().coordinate_ring()
P = []
for i in range(len(self._fastpolys[0])):
# Check if denominator is the identity;
#if not, then must append the fraction evaluated at the point
if self._fastpolys[1][i] is R.one():
P.append(self._fastpolys[0][i](*x))
else:
P.append(self._fastpolys[0][i](*x)/self._fastpolys[1][i](*x))
return P
def homogenize(self, n):
r"""
Return the homogenization of this map.
If it's domain is a subscheme, the domain of
the homogenized map is the projective embedding of the domain. The domain and codomain
can be homogenized at different coordinates: ``n[0]`` for the domain and ``n[1]`` for the codomain.
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:`SchemMorphism_polynomial_projective_space`.
EXAMPLES::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: H = Hom(A, A)
sage: f = H([(x^2-2)/x^5, y^2])
sage: f.homogenize(2)
Scheme endomorphism of Projective Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(x0^2*x2^5 - 2*x2^7 : x0^5*x1^2 : x0^5*x2^2)
::
sage: A.<x,y> = AffineSpace(CC, 2)
sage: H = Hom(A, A)
sage: f = H([(x^2-2)/(x*y), y^2-x])
sage: f.homogenize((2, 0))
Scheme morphism:
From: Projective Space of dimension 2 over Complex Field with 53 bits of precision
To: Projective Space of dimension 2 over Complex Field with 53 bits of precision
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(x0*x1*x2^2 : x0^2*x2^2 + (-2.00000000000000)*x2^4 : x0*x1^3 - x0^2*x1*x2)
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: X = A.subscheme([x-y^2])
sage: H = Hom(X, X)
sage: f = H([9*y^2, 3*y])
sage: f.homogenize(2)
Scheme endomorphism of Closed subscheme of Projective Space
of dimension 2 over Integer Ring defined by:
x1^2 - x0*x2
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(9*x1^2 : 3*x1*x2 : x2^2)
::
sage: R.<t> = PolynomialRing(ZZ)
sage: A.<x,y> = AffineSpace(R, 2)
sage: H = Hom(A, A)
sage: f = H([(x^2-2)/y, y^2-x])
sage: f.homogenize((2, 0))
Scheme morphism:
From: Projective Space of dimension 2 over Univariate Polynomial Ring in t over Integer Ring
To: Projective Space of dimension 2 over Univariate Polynomial Ring in t over Integer Ring
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(x1*x2^2 : x0^2*x2 + (-2)*x2^3 : x1^3 - x0*x1*x2)
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: H = End(A)
sage: f = H([x^2-1])
sage: f.homogenize((1, 0))
Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x0 : x1) to
(x1^2 : x0^2 - x1^2)
::
R.<a> = PolynomialRing(QQbar)
A.<x,y> = AffineSpace(R, 2)
H = End(A)
f = H([QQbar(sqrt(2))*x*y, a*x^2])
f.homogenize(2)
Scheme endomorphism of Projective Space of dimension 2 over Univariate
Polynomial Ring in a over Algebraic Field
Defn: Defined on coordinates by sending (x0 : x1 : x2) to
(1.414213562373095?*x0*x1 : a*x0^2 : x2^2)
::
sage: P.<x,y,z> = AffineSpace(QQ, 3)
sage: H = End(P)
sage: f = H([x^2 - 2*x*y + z*x, z^2 -y^2 , 5*z*y])
sage: f.homogenize(2).dehomogenize(2) == f
True
::
sage: K.<c> = FunctionField(QQ)
sage: A.<x> = AffineSpace(K, 1)
sage: f = Hom(A, A)([x^2 + c])
sage: f.homogenize(1)
Scheme endomorphism of Projective Space of
dimension 1 over Rational function field in c over Rational Field
Defn: Defined on coordinates by sending (x0 : x1) to
(x0^2 + c*x1^2 : x1^2)
::
sage: A.<z> = AffineSpace(QQbar, 1)
sage: H = End(A)
sage: f = H([2*z / (z^2+2*z+3)])
sage: f.homogenize(1)
Scheme endomorphism of Projective Space of dimension 1 over Algebraic
Field
Defn: Defined on coordinates by sending (x0 : x1) to
(x0*x1 : 1/2*x0^2 + x0*x1 + 3/2*x1^2)
"""
#it is possible to homogenize the domain and codomain at different coordinates
if isinstance(n, (tuple, list)):
ind = tuple(n)
else:
ind = (n, n)
#homogenize the domain and codomain
A = self.domain().projective_embedding(ind[0]).codomain()
B = self.codomain().projective_embedding(ind[1]).codomain()
H = Hom(A, B)
newvar = A.ambient_space().coordinate_ring().gen(ind[0])
N = A.ambient_space().dimension_relative()
M = B.ambient_space().dimension_relative()
#create dictionary for mapping of coordinate rings
R = self.domain().ambient_space().coordinate_ring()
S = A.ambient_space().coordinate_ring()
Rvars = R.gens()
vars = list(S.gens())
vars.remove(S.gen(ind[0]))
D = dict([[Rvars[i],vars[i]] for i in range(N)])
#clear the denominators if a rational function
L = [self[i].denominator() for i in range(M)]
l = [prod(L[:j] + L[j+1:M]) for j in range(M)]
F = [S(R(self[i].numerator()*l[i]).subs(D)) for i in range(M)]
#homogenize
F.insert(ind[1], S(R(prod(L)).subs(D))) #coerce in case l is a constant
try:
#remove possible gcd of the polynomials
g = gcd(F)
F = [S(f/g) for f in F]
#remove possible gcd of coefficients
gc = gcd([f.content() for f in F])
F = [S(f/gc) for f in F]
except (AttributeError, ValueError): #no gcd
pass
d = max([F[i].degree() for i in range(M+1)])
F = [F[i].homogenize(str(newvar))*newvar**(d-F[i].degree()) for i in range(M+1)]
return(H(F))
def dynatomic_polynomial(self, period):
r"""
For a map `f:\mathbb{A}^1 \to \mathbb{A}^1` this function computes
the (affine) dynatomic polynomial.
The dynatomic polynomial is the analog of the cyclotomic polynomial and its roots are the points
of formal period `n`.
ALGORITHM:
Homogenize to a map `f:\mathbb{P}^1 \to \mathbb{P}^1` and compute the dynatomic polynomial there.
Then, dehomogenize.
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 single variable polynomial in the coordinate ring of the polynomial. \
Otherwise a fraction field element of the coordinate ring of the polynomial.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: H = Hom(A, A)
sage: f = H([x^2+y^2, y^2])
sage: f.dynatomic_polynomial(2)
Traceback (most recent call last):
...
TypeError: does not make sense in dimension >1
::
sage: A.<x> = AffineSpace(ZZ, 1)
sage: H = Hom(A, A)
sage: f = H([(x^2+1)/x])
sage: f.dynatomic_polynomial(4)
2*x^12 + 18*x^10 + 57*x^8 + 79*x^6 + 48*x^4 + 12*x^2 + 1
::
sage: A.<x> = AffineSpace(CC, 1)
sage: H = Hom(A, A)
sage: f = H([(x^2+1)/(3*x)])
sage: f.dynatomic_polynomial(3)
13.0000000000000*x^6 + 117.000000000000*x^4 + 78.0000000000000*x^2 +
1.00000000000000
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: H = Hom(A, A)
sage: f = H([x^2-10/9])
sage: f.dynatomic_polynomial([2, 1])
531441*x^4 - 649539*x^2 - 524880
::
sage: A.<x> = AffineSpace(CC, 1)
sage: H = Hom(A, A)
sage: f = H([x^2+CC.0])
sage: f.dynatomic_polynomial(2)
x^2 + x + 1.00000000000000 + 1.00000000000000*I
::
sage: K.<c> = FunctionField(QQ)
sage: A.<x> = AffineSpace(K, 1)
sage: f = Hom(A, A)([x^2 + c])
sage: f.dynatomic_polynomial(4)
x^12 + 6*c*x^10 + x^9 + (15*c^2 + 3*c)*x^8 + 4*c*x^7 + (20*c^3 + 12*c^2 + 1)*x^6
+ (6*c^2 + 2*c)*x^5 + (15*c^4 + 18*c^3 + 3*c^2 + 4*c)*x^4 + (4*c^3 + 4*c^2 + 1)*x^3
+ (6*c^5 + 12*c^4 + 6*c^3 + 5*c^2 + c)*x^2 + (c^4 + 2*c^3 + c^2 + 2*c)*x
+ c^6 + 3*c^5 + 3*c^4 + 3*c^3 + 2*c^2 + 1
::
sage: A.<z> = AffineSpace(QQ, 1)
sage: H = End(A)
sage: f = H([z^2+3/z+1/7])
sage: f.dynatomic_polynomial(1).parent()
Multivariate Polynomial Ring in z over Rational Field
"""
if self.domain() != self.codomain():
raise TypeError("must have same domain and codomain to iterate")
from sage.schemes.affine.affine_space import is_AffineSpace
if is_AffineSpace(self.domain())==False:
raise NotImplementedError("not implemented for subschemes")
if self.domain().dimension_relative()>1:
raise TypeError("does not make sense in dimension >1")
F = self.homogenize(1).dynatomic_polynomial(period)
S = self.domain().coordinate_ring()
if S(F.denominator()).degree() == 0:
R = F.parent()
phi = R.hom([S.gen(0), 1], S)
return(phi(F))
else:
R = F.numerator().parent()
phi = R.hom([S.gen(0), 1], S)
return(phi(F.numerator())/phi(F.denominator()))
def nth_iterate_map(self, n):
r"""
This function returns the ``n``-th iterate of the map.
ALGORITHM:
Uses a form of successive squaring to reducing computations.
.. TODO::
This could be improved.
INPUT:
- ``n`` - a positive integer.
OUTPUT:
- A map between Affine spaces.
EXAMPLES::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: H = Hom(A, A)
sage: f = H([(x^2-2)/(2*y), y^2-3*x])
sage: f.nth_iterate_map(2)
Scheme endomorphism of Affine Space of dimension 2 over Integer Ring
Defn: Defined on coordinates by sending (x, y) to
((x^4 - 4*x^2 - 8*y^2 + 4)/(8*y^4 - 24*x*y^2), (2*y^5 - 12*x*y^3
+ 18*x^2*y - 3*x^2 + 6)/(2*y))
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: H = Hom(A, A)
sage: f = H([(3*x^2-2)/(x)])
sage: f.nth_iterate_map(3)
Scheme endomorphism of Affine Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x) to
((2187*x^8 - 6174*x^6 + 6300*x^4 - 2744*x^2 + 432)/(81*x^7 -
168*x^5 + 112*x^3 - 24*x))
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: X = A.subscheme([x-y^2])
sage: H = Hom(X, X)
sage: f = H([9*x^2, 3*y])
sage: f.nth_iterate_map(2)
Scheme endomorphism of Closed subscheme of Affine Space of dimension 2
over Integer Ring defined by:
-y^2 + x
Defn: Defined on coordinates by sending (x, y) to
(729*x^4, 9*y)
"""
if self.domain() != self.codomain():
raise TypeError("domain and codomain of function not equal")
N = self.codomain().ambient_space().dimension_relative()
F = list(self._polys)
R = F[0].parent()
Coord_ring = self.codomain().ambient_space().coordinate_ring()
D = Integer(n).digits(2)
PHI = [Coord_ring.gen(i) for i in range(N)]
for i in range(len(D)):
T = [F[j] for j in range(N)]
for k in range(D[i]):
PHI = [PHI[j](T) for j in range(N)]
if i != len(D)-1: #avoid extra iterate
F = [R(F[j](T)) for j in range(N)] #'square'
H = Hom(self.domain(), self.codomain())
return(H(PHI))
def nth_iterate(self, P, n):
r"""
Returns the ``n``-th iterate of the point ``P`` by this map.
INPUT:
- ``P`` -- a point in the map's domain.
- ``n`` -- a positive integer.
OUTPUT:
- a point in the map's codomain.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: H = Hom(A, A)
sage: f = H([(x-2*y^2)/x, 3*x*y])
sage: f.nth_iterate(A(9, 3), 3)
(-104975/13123, -9566667)
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: X = A.subscheme([x-y^2])
sage: H = Hom(X, X)
sage: f = H([9*y^2, 3*y])
sage: f.nth_iterate(X(9, 3), 4)
(59049, 243)
::
sage: R.<t> = PolynomialRing(QQ)
sage: A.<x,y> = AffineSpace(FractionField(R), 2)
sage: H = Hom(A, A)
sage: f = H([(x-t*y^2)/x, t*x*y])
sage: f.nth_iterate(A(1, t), 3)
((-t^16 + 3*t^13 - 3*t^10 + t^7 + t^5 + t^3 - 1)/(t^5 + t^3 - 1), -t^9 - t^7 + t^4)
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: X = A.subscheme([x^2-y^2])
sage: H = End(X)
sage: u,v,w = X.coordinate_ring().gens()
sage: f = H([x^2, y^2, x+y])
sage: f.nth_iterate_map(2)
Scheme endomorphism of Closed subscheme of Affine Space of dimension 3
over Rational Field defined by:
x^2 - y^2
Defn: Defined on coordinates by sending (x, y, z) to
(x^4, y^4, x^2 + y^2)
"""
return(P.nth_iterate(self, n))
def orbit(self, P, n):
r"""
Returns the orbit of ``P`` by the map.
If `n` is an integer it returns `[P,self(P),\ldots,self^n(P)]`.
If `n` is a list or tuple `n=[m,k]` it returns `[self^m(P),\ldots,self^k(P)]`.
INPUT:
- ``P`` -- a point in the map's domain.
- ``n`` -- a non-negative integer or list or tuple of two non-negative integers.
OUTPUT:
- a list of points in the map's codomain.
EXAMPLES::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: H = Hom(A, A)
sage: f = H([(x-2*y^2)/x, 3*x*y])
sage: f.orbit(A(9, 3), 3)
[(9, 3), (-1, 81), (13123, -243), (-104975/13123, -9566667)]
::
sage: A.<x> = AffineSpace(QQ, 1)
sage: H = Hom(A, A)
sage: f = H([(x-2)/x])
sage: f.orbit(A(1/2), [1, 3])
[(-3), (5/3), (-1/5)]
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: X = A.subscheme([x-y^2])
sage: H = Hom(X, X)
sage: f = H([9*y^2, 3*y])
sage: f.orbit(X(9, 3), (0, 4))
[(9, 3), (81, 9), (729, 27), (6561, 81), (59049, 243)]
::
sage: R.<t> = PolynomialRing(QQ)
sage: A.<x,y> = AffineSpace(FractionField(R), 2)
sage: H = Hom(A, A)
sage: f = H([(x-t*y^2)/x, t*x*y])
sage: f.orbit(A(1, t), 3)
[(1, t), (-t^3 + 1, t^2), ((-t^5 - t^3 + 1)/(-t^3 + 1), -t^6 + t^3),
((-t^16 + 3*t^13 - 3*t^10 + t^7 + t^5 + t^3 - 1)/(t^5 + t^3 - 1), -t^9 -
t^7 + t^4)]
"""
return(P.orbit(self, n))
def global_height(self, prec=None):
r"""
Returns the maximum of the heights of the coefficients in any
of the coordinate functions of the affine morphism.
INPUT:
- ``prec`` -- desired floating point precision (default:
default RealField precision).
OUTPUT: A real number.
EXAMPLES::
sage: A.<x> = AffineSpace(QQ, 1)
sage: H = Hom(A, A)
sage: f = H([1/1331*x^2 + 4000]);
sage: f.global_height()
8.29404964010203
::
sage: R.<x> = PolynomialRing(QQ)
sage: k.<w> = NumberField(x^2 + 5)
sage: A.<x,y> = AffineSpace(k, 2)
sage: H = Hom(A, A)
sage: f = H([13*w*x^2 + 4*y, 1/w*y^2]);
sage: f.global_height(prec=100)
3.3696683136785869233538671082
::
sage: A.<x> = AffineSpace(ZZ, 1)
sage: H = Hom(A, A)
sage: f = H([7*x^2 + 1513]);
sage: f.global_height()
7.32184971378836
"""
H=0
for i in range(self.domain().ambient_space().dimension_relative()):
C = self[i].coefficients()
if C == []: #to deal with the case self[i]=0
h=0
else:
h = max([c.global_height(prec) for c in C])
H = max(H,h)
return(H)
def jacobian (self):
r"""
Returns the Jacobian matrix of partial derivitive of this map.
The `(i, j)` entry of the Jacobian matrix is the partial derivative
`diff(functions[i], variables[j])`.
OUTPUT:
- matrix with coordinates in the coordinate ring of the map.
EXAMPLES::
sage: A.<z> = AffineSpace(QQ, 1)
sage: H = End(A)
sage: f = H([z^2 - 3/4])
sage: f.jacobian()
[2*z]
::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: H = End(A)
sage: f = H([x^3 - 25*x + 12*y, 5*y^2*x - 53*y + 24])
sage: f.jacobian()
[ 3*x^2 - 25 12]
[ 5*y^2 10*x*y - 53]
::
sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: H = End(A)
sage: f = H([(x^2 - x*y)/(1+y), (5+y)/(2+x)])
sage: f.jacobian()
[ (2*x - y)/(y + 1) (-x^2 - x)/(y^2 + 2*y + 1)]
[ (-y - 5)/(x^2 + 4*x + 4) 1/(x + 2)]
"""
try:
return self.__jacobian
except AttributeError:
pass
self.__jacobian = jacobian(list(self),self.domain().ambient_space().gens())
return self.__jacobian
def multiplier(self, P, n, check=True):
r"""
Returns the multiplier of the point ``P`` of period ``n`` by the map.
The map must be an endomorphism.
INPUT:
- ``P`` - a point on domain of the map.
- ``n`` - a positive integer, the period of ``P``.
- ``check`` -- verify that ``P`` has period ``n``, Default:True.
OUTPUT:
- a square matrix of size ``self.codomain().dimension_relative()`` in
the ``base_ring`` of the map.
EXAMPLES::
sage: P.<x,y> = AffineSpace(QQ, 2)
sage: H = End(P)
sage: f = H([x^2, y^2])
sage: f.multiplier(P([1, 1]), 1)
[2 0]
[0 2]
::
sage: P.<x,y,z> = AffineSpace(QQ, 3)
sage: H = End(P)
sage: f = H([x, y^2, z^2 - y])
sage: f.multiplier(P([1/2, 1, 0]), 2)
[1 0 0]
[0 4 0]
[0 0 0]
::
sage: P.<x> = AffineSpace(CC, 1)
sage: H = End(P)
sage: f = H([x^2 + 1/2])
sage: f.multiplier(P([0.5 + 0.5*I]), 1)
[1.00000000000000 + 1.00000000000000*I]
::
sage: R.<t> = PolynomialRing(CC, 1)
sage: P.<x> = AffineSpace(R, 1)
sage: H = End(P)
sage: f = H([x^2 - t^2 + t])
sage: f.multiplier(P([-t + 1]), 1)
[(-2.00000000000000)*t + 2.00000000000000]
::
sage: P.<x,y> = AffineSpace(QQ, 2)
sage: X = P.subscheme([x^2-y^2])
sage: H = End(X)
sage: f = H([x^2, y^2])
sage: f.multiplier(X([1, 1]), 1)
[2 0]
[0 2]
"""
if not self.is_endomorphism():
raise TypeError("must be an endomorphism")
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