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r"""

Projective `n` space over a ring

EXAMPLES:

We construct projective space over various rings of various dimensions.

The simplest projective space::

sage: ProjectiveSpace(0)

Projective Space of dimension 0 over Integer Ring

A slightly bigger projective space over `\QQ`::

sage: X = ProjectiveSpace(1000, QQ); X

Projective Space of dimension 1000 over Rational Field

sage: X.dimension()

1000

We can use "over" notation to create projective spaces over various

base rings.

sage: X = ProjectiveSpace(5)/QQ; X

Projective Space of dimension 5 over Rational Field

sage: X/CC

Projective Space of dimension 5 over Complex Field with 53 bits of precision

The third argument specifies the printing names of the generators of the

homogenous coordinate ring. Using the method `.objgens()` you can obtain both

the space and the generators as ready to use variables. ::

sage: P2, vars = ProjectiveSpace(10, QQ, 't').objgens()

sage: vars

(t0, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10)

You can alternatively use the special syntax with ``<`` and ``>``.

sage: P2.<x,y,z> = ProjectiveSpace(2, QQ)

sage: P2

Projective Space of dimension 2 over Rational Field

sage: P2.coordinate_ring()

Multivariate Polynomial Ring in x, y, z over Rational Field

The first of the three lines above is just equivalent to the two lines::

sage: P2 = ProjectiveSpace(2, QQ, 'xyz')

sage: x,y,z = P2.gens()

For example, we use `x,y,z` to define the intersection of

two lines.

sage: V = P2.subscheme([x+y+z, x+y-z]); V

Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:

x + y + z,

x + y - z

sage: V.dimension()

AUTHORS:

- Ben Hutz: (June 2012): support for rings

- Ben Hutz (9/2014): added support for Cartesian products

- Rebecca Lauren Miller (March 2016) : added point_transformation_matrix

"""

#*****************************************************************************

# Distributed under the terms of the GNU General Public License (GPL)

# http://www.gnu.org/licenses/

#*****************************************************************************

from __future__ import print_function

from six.moves import range

from six import integer_types

from sage.arith.misc import binomial

from sage.rings.all import (PolynomialRing,

Integer,

ZZ)

from sage.rings.ring import CommutativeRing

from sage.rings.rational_field import is_RationalField

from sage.rings.polynomial.multi_polynomial_ring import is_MPolynomialRing

from sage.rings.finite_rings.finite_field_constructor import is_FiniteField

from sage.categories.fields import Fields

_Fields = Fields()

from sage.categories.homset import Hom, End

from sage.categories.number_fields import NumberFields

from sage.categories.map import Map

from sage.misc.all import (latex,

prod)

from sage.structure.category_object import normalize_names

from sage.arith.all import gcd, binomial

from sage.combinat.integer_vector import IntegerVectors

from sage.combinat.integer_vector_weighted import WeightedIntegerVectors

from sage.combinat.permutation import Permutation

from sage.combinat.tuple import Tuples

from sage.combinat.tuple import UnorderedTuples

from sage.matrix.constructor import matrix

from sage.modules.free_module_element import prepare

from sage.schemes.generic.ambient_space import AmbientSpace

from sage.schemes.projective.projective_homset import (SchemeHomset_points_projective_ring,

SchemeHomset_points_projective_field)

from sage.schemes.projective.projective_point import (SchemeMorphism_point_projective_ring,

SchemeMorphism_point_projective_field,

SchemeMorphism_point_projective_finite_field)

from sage.schemes.projective.projective_morphism import (SchemeMorphism_polynomial_projective_space,

SchemeMorphism_polynomial_projective_space_field,

SchemeMorphism_polynomial_projective_space_finite_field)

def is_ProjectiveSpace(x):

r"""

Return True if ``x`` is a projective space.

In other words, if ``x`` is an ambient space `\mathbb{P}^n_R`,

where `R` is a ring and `n\geq 0` is an integer.

EXAMPLES::

sage: from sage.schemes.projective.projective_space import is_ProjectiveSpace

sage: is_ProjectiveSpace(ProjectiveSpace(5, names='x'))

True

sage: is_ProjectiveSpace(ProjectiveSpace(5, GF(9,'alpha'), names='x'))

True

sage: is_ProjectiveSpace(Spec(ZZ))

False

"""

return isinstance(x, ProjectiveSpace_ring)

def ProjectiveSpace(n, R=None, names='x'):

r"""

Return projective space of dimension ``n`` over the ring ``R``.

EXAMPLES: The dimension and ring can be given in either order.

sage: ProjectiveSpace(3, QQ)

Projective Space of dimension 3 over Rational Field

sage: ProjectiveSpace(5, QQ)

Projective Space of dimension 5 over Rational Field

sage: P = ProjectiveSpace(2, QQ, names='XYZ'); P

Projective Space of dimension 2 over Rational Field

sage: P.coordinate_ring()

Multivariate Polynomial Ring in X, Y, Z over Rational Field

The divide operator does base extension.

sage: ProjectiveSpace(5)/GF(17)

Projective Space of dimension 5 over Finite Field of size 17

The default base ring is `\ZZ`.

sage: ProjectiveSpace(5)

Projective Space of dimension 5 over Integer Ring

There is also an projective space associated each polynomial ring.

sage: R = GF(7)['x,y,z']

sage: P = ProjectiveSpace(R); P

Projective Space of dimension 2 over Finite Field of size 7

sage: P.coordinate_ring()

Multivariate Polynomial Ring in x, y, z over Finite Field of size 7

sage: P.coordinate_ring() is R

True

sage: ProjectiveSpace(3, Zp(5), 'y')

Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20

sage: ProjectiveSpace(2,QQ,'x,y,z')

Projective Space of dimension 2 over Rational Field

sage: PS.<x,y>=ProjectiveSpace(1,CC)

sage: PS

Projective Space of dimension 1 over Complex Field with 53 bits of precision

sage: R.<x,y,z> = QQ[]

sage: ProjectiveSpace(R).variable_names()

('x', 'y', 'z')

Projective spaces are not cached, i.e., there can be several with

the same base ring and dimension (to facilitate gluing

constructions).

"""

if is_MPolynomialRing(n) and R is None:

A = ProjectiveSpace(n.ngens()-1, n.base_ring(), names=n.variable_names())

A._coordinate_ring = n

return A

if isinstance(R, integer_types + (Integer,)):

n, R = R, n

if R is None:

R = ZZ # default is the integers

if R in _Fields:

if is_FiniteField(R):

return ProjectiveSpace_finite_field(n, R, names)

if is_RationalField(R):

return ProjectiveSpace_rational_field(n, R, names)

else:

return ProjectiveSpace_field(n, R, names)

elif isinstance(R, CommutativeRing):

return ProjectiveSpace_ring(n, R, names)

else:

raise TypeError("R (=%s) must be a commutative ring"%R)

class ProjectiveSpace_ring(AmbientSpace):

"""

Projective space of dimension `n` over the ring

`R`.

EXAMPLES::

sage: X.<x,y,z,w> = ProjectiveSpace(3, QQ)

sage: X.base_scheme()

Spectrum of Rational Field

sage: X.base_ring()

Rational Field

sage: X.structure_morphism()

Scheme morphism:

From: Projective Space of dimension 3 over Rational Field

To: Spectrum of Rational Field

Defn: Structure map

sage: X.coordinate_ring()

Multivariate Polynomial Ring in x, y, z, w over Rational Field

Loading and saving::

sage: loads(X.dumps()) == X

True

"""

def __init__(self, n, R=ZZ, names=None):

"""

Initialization function.

EXAMPLES::

sage: ProjectiveSpace(3, Zp(5), 'y')

Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20

"""

names = normalize_names(n+1, names)

AmbientSpace.__init__(self, n, R)

self._assign_names(names)

def ngens(self):

"""

Return the number of generators of this projective space.

This is the number of variables in the coordinate ring of self.

EXAMPLES::

sage: ProjectiveSpace(3, QQ).ngens()

sage: ProjectiveSpace(7, ZZ).ngens()

"""

return self.dimension_relative() + 1

def _check_satisfies_equations(self, v):

"""

Return True if ``v`` defines a point on the scheme; raise a

TypeError otherwise.

EXAMPLES::

sage: P = ProjectiveSpace(2, ZZ)

sage: P._check_satisfies_equations([1, 1, 0])

True

sage: P = ProjectiveSpace(1, QQ)

sage: P._check_satisfies_equations((1/2, 0))

True

sage: P = ProjectiveSpace(2, ZZ)

sage: P._check_satisfies_equations([0, 0, 0])

Traceback (most recent call last):

...

TypeError: the zero vector is not a point in projective space

sage: P = ProjectiveSpace(2, ZZ)

sage: P._check_satisfies_equations((1, 0))

Traceback (most recent call last):

...

TypeError: the list v=(1, 0) must have 3 components

sage: P = ProjectiveSpace(2, ZZ)

sage: P._check_satisfies_equations([1/2, 0, 1])

Traceback (most recent call last):

...

TypeError: the components of v=[1/2, 0, 1] must be elements of Integer Ring

"""

if not isinstance(v, (list, tuple)):

raise TypeError('the argument v=%s must be a list or tuple'%v)

n = self.ngens()

if not len(v) == n:

raise TypeError('the list v=%s must have %s components'%(v, n))

R = self.base_ring()

for coord in v:

if not coord in R:

raise TypeError('the components of v=%s must be elements of %s'%(v, R))

zero = [R(0)]*n

if v == zero:

raise TypeError('the zero vector is not a point in projective space')

return True

def coordinate_ring(self):

"""

Return the coordinate ring of this scheme.

EXAMPLES::

sage: ProjectiveSpace(3, GF(19^2,'alpha'), 'abcd').coordinate_ring()

Multivariate Polynomial Ring in a, b, c, d over Finite Field in alpha of size 19^2

sage: ProjectiveSpace(3).coordinate_ring()

Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring

sage: ProjectiveSpace(2, QQ, ['alpha', 'beta', 'gamma']).coordinate_ring()

Multivariate Polynomial Ring in alpha, beta, gamma over Rational Field

"""

try:

return self._coordinate_ring

except AttributeError:

self._coordinate_ring = PolynomialRing(self.base_ring(),

self.variable_names(), self.dimension_relative()+1)

return self._coordinate_ring

def _validate(self, polynomials):

"""

If ``polynomials`` is a tuple of valid polynomial functions on self,

return ``polynomials``, otherwise raise TypeError.

Since this is a projective space, polynomials must be homogeneous.

INPUT:

- ``polynomials`` -- tuple of polynomials in the coordinate ring of

this space.

OUTPUT:

- tuple of polynomials in the coordinate ring of this space.

EXAMPLES::

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)

sage: P._validate([x*y - z^2, x])

[x*y - z^2, x]

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)

sage: P._validate((x*y - z, x))

Traceback (most recent call last):

...

TypeError: x*y - z is not a homogeneous polynomial

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)

sage: P._validate(x*y - z)

Traceback (most recent call last):

...

TypeError: the argument polynomials=x*y - z must be a list or tuple

"""

if not isinstance(polynomials, (list, tuple)):

raise TypeError('the argument polynomials=%s must be a list or tuple'%polynomials)

for f in polynomials:

if not f.is_homogeneous():

raise TypeError("%s is not a homogeneous polynomial" % f)

return polynomials

def __eq__(self, right):

"""

Check equality of two projective spaces.

EXAMPLES::

sage: ProjectiveSpace(QQ, 3, 'a') == ProjectiveSpace(ZZ, 3, 'a')

False

sage: ProjectiveSpace(ZZ, 1, 'a') == ProjectiveSpace(ZZ, 0, 'a')

False

sage: ProjectiveSpace(ZZ, 2, 'a') == AffineSpace(ZZ, 2, 'a')

False

sage: P = ProjectiveSpace(ZZ, 1, 'x')

sage: loads(P.dumps()) == P

True

"""

if not isinstance(right, ProjectiveSpace_ring):

return False

return (self.dimension_relative() == right.dimension_relative() and

self.coordinate_ring() == right.coordinate_ring())

def __ne__(self, other):

"""

Check non-equality of two projective spaces.

EXAMPLES::

sage: ProjectiveSpace(QQ, 3, 'a') != ProjectiveSpace(ZZ, 3, 'a')

True

sage: ProjectiveSpace(ZZ, 1, 'a') != ProjectiveSpace(ZZ, 0, 'a')

True

sage: ProjectiveSpace(ZZ, 2, 'a') != AffineSpace(ZZ, 2, 'a')

True

"""

return not (self == other)

def __pow__(self, m):

"""

Return the Cartesian power of this space.

INPUT: ``m`` -- integer.

OUTPUT: product of projective spaces.

EXAMPLES::

sage: P = ProjectiveSpace(1, QQ, 'x')

sage: P3 = P^3; P3

Product of projective spaces P^1 x P^1 x P^1 over Rational Field

sage: P3.variable_names()

('x0', 'x1', 'x2', 'x3', 'x4', 'x5')

As you see, custom variable names are not preserved by power operator,

since there is no natural way to make new ones in general.

"""

mm = int(m)

if mm != m:

raise ValueError("m must be an integer")

from sage.schemes.product_projective.space import ProductProjectiveSpaces

return ProductProjectiveSpaces([self.dimension_relative()] * mm, self.base_ring())

def __mul__(self, right):

r"""

Create the product of projective spaces.

INPUT:

- ``right`` - a projective space, product of projective spaces, or subscheme.

OUTPUT: a product of projective spaces or subscheme.

EXAMPLES::

sage: P1 = ProjectiveSpace(QQ, 1, 'x')

sage: P2 = ProjectiveSpace(QQ, 2, 'y')

sage: P1*P2

Product of projective spaces P^1 x P^2 over Rational Field

sage: S.<t,x,y,z,u,v,w> = ProductProjectiveSpaces([3, 2], QQ)

sage: T.<a,b> = ProjectiveSpace(QQ, 1)

sage: T*S

Product of projective spaces P^1 x P^3 x P^2 over Rational Field

sage: S = ProjectiveSpace(ZZ, 2, 't')

sage: T = ProjectiveSpace(ZZ, 3, 'x')

sage: T.inject_variables()

Defining x0, x1, x2, x3

sage: X = T.subscheme([x0*x2 - x1*x3])

sage: S*X

Closed subscheme of Product of projective spaces P^2 x P^3 over Integer Ring defined by:

x0*x2 - x1*x3

sage: S = ProjectiveSpace(QQ, 3, 'x')

sage: T = AffineSpace(2, QQ, 'y')

sage: S*T

Traceback (most recent call last):

...

TypeError: Affine Space of dimension 2 over Rational Field must be a

projective space, product of projective spaces, or subscheme

"""

if self.base_ring() != right.base_ring():

raise ValueError ('Must have the same base ring')

from sage.schemes.product_projective.space import ProductProjectiveSpaces_ring

from sage.schemes.product_projective.space import ProductProjectiveSpaces

from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme

if isinstance(right, (ProductProjectiveSpaces_ring)):

return ProductProjectiveSpaces([self] + right.components())

elif isinstance(right, ProjectiveSpace_ring):

if self is right:

return self.__pow__(2)

return ProductProjectiveSpaces([self, right])

elif isinstance(right, AlgebraicScheme_subscheme):

AS = self*right.ambient_space()

CR = AS.coordinate_ring()

n = self.ambient_space().coordinate_ring().ngens()

phi = self.ambient_space().coordinate_ring().hom(list(CR.gens()[:n]), CR)

psi = right.ambient_space().coordinate_ring().hom(list(CR.gens()[n:]), CR)

return AS.subscheme([phi(t) for t in self.defining_polynomials()] + [psi(t) for t in right.defining_polynomials()])

else:

raise TypeError('%s must be a projective space, product of projective spaces, or subscheme'%right)

def _latex_(self):

r"""

Return a LaTeX representation of this projective space.

EXAMPLES::

sage: print(latex(ProjectiveSpace(1, ZZ, 'x')))

{\mathbf P}_{\Bold{Z}}^1

TESTS::

sage: ProjectiveSpace(3, Zp(5), 'y')._latex_()

'{\\mathbf P}_{\\ZZ_{5}}^3'

"""

return "{\\mathbf P}_{%s}^%s"%(latex(self.base_ring()), self.dimension_relative())

def _linear_system_as_kernel(self, d, pt, m):

"""

Return a matrix whose kernel consists of the coefficient vectors

of the degree ``d`` hypersurfaces (wrt lexicographic ordering of its

monomials) with multiplicity at least ``m`` at ``pt``.

INPUT:

- ``d`` -- a nonnegative integer.

- ``pt`` -- a point of self (possibly represented by a list with at \

least one component equal to 1).

- ``m`` -- a nonnegative integer.

OUTPUT:

A matrix of size `\binom{m-1+n}{n}` x `\binom{d+n}{n}` where n is the

relative dimension of self. The base ring of the matrix is a ring that

contains the base ring of self and the coefficients of the given point.

EXAMPLES:

If the degree `d` is 0, then a matrix consisting of the first unit vector

is returned::

sage: P = ProjectiveSpace(GF(5), 2, names='x')

sage: pt = P([1, 1, 1])

sage: P._linear_system_as_kernel(0, pt, 3)

[1]

[0]

If the multiplicity `m` is 0, then the a matrix with zero rows

is returned::

sage: P = ProjectiveSpace(GF(5), 2, names='x')

sage: pt = P([1, 1, 1])

sage: M = P._linear_system_as_kernel(2, pt, 0)

sage: [M.nrows(), M.ncols()]

[0, 6]

The base ring does not need to be a field or even an integral domain.

In this case, the point can be given by a list::

sage: R = Zmod(4)

sage: P = ProjectiveSpace(R, 2, names='x')

sage: pt = [R(1), R(3), R(0)]

sage: P._linear_system_as_kernel(3, pt, 2)

[1 3 0 1 0 0 3 0 0 0]

[0 1 0 2 0 0 3 0 0 0]

[0 0 1 0 3 0 0 1 0 0]

When representing a point by a list at least one component must be 1

(even when the base ring is a field and the list gives a well-defined

point in projective space)::

sage: R = GF(5)

sage: P = ProjectiveSpace(R, 2, names='x')

sage: pt = [R(3), R(3), R(0)]

sage: P._linear_system_as_kernel(3, pt, 2)

Traceback (most recent call last):

...

TypeError: at least one component of pt=[3, 3, 0] must be equal

to 1

The components of the list do not have to be elements of the base ring

of the projective space. It suffices if there exists a common parent.

For example, the kernel of the following matrix corresponds to

hypersurfaces of degree 2 in 3-space with multiplicity at least 2 at a

general point in the third affine patch::

sage: P = ProjectiveSpace(QQ,3,names='x')

sage: RPol.<t0,t1,t2,t3> = PolynomialRing(QQ,4)

sage: pt = [t0,t1,1,t3]

sage: P._linear_system_as_kernel(2,pt,2)

[ 2*t0 t1 1 t3 0 0 0 0 0 0]

[ 0 t0 0 0 2*t1 1 t3 0 0 0]

[ t0^2 t0*t1 t0 t0*t3 t1^2 t1 t1*t3 1 t3 t3^2]

[ 0 0 0 t0 0 0 t1 0 1 2*t3]

.. TODO::

Use this method as starting point to implement a class

LinearSystem for linear systems of hypersurfaces.

"""

if not isinstance(d, (int, Integer)):

raise TypeError('the argument d=%s must be an integer'%d)

if d < 0:

raise ValueError('the integer d=%s must be nonnegative'%d)

if not isinstance(pt, (list, tuple, \

SchemeMorphism_point_projective_ring)):

raise TypeError('the argument pt=%s must be a list, tuple, or '

'point on a projective space'%pt)

pt, R = prepare(pt, None)

n = self.dimension_relative()

if not len(pt) == n+1:

raise TypeError('the sequence pt=%s must have %s '

'components'%(pt, n + 1))

if not R.has_coerce_map_from(self.base_ring()):

raise TypeError('unable to find a common ring for all elements')

try:

i = pt.index(1)

except Exception:

raise TypeError('at least one component of pt=%s must be equal '

'to 1'%pt)

pt = pt[:i] + pt[i+1:]

if not isinstance(m, (int, Integer)):

raise TypeError('the argument m=%s must be an integer'%m)

if m < 0:

raise ValueError('the integer m=%s must be nonnegative'%m)

# the components of partials correspond to partial derivatives

# of order at most m-1 with respect to n variables

partials = IntegerVectors(m-1, n+1).list()

# the components of monoms correspond to monomials of degree

# at most d in n variables

monoms = IntegerVectors(d, n+1).list()

M = matrix(R,len(partials),len(monoms))

for row in range(M.nrows()):

e = partials[row][:i] + partials[row][i+1:]

for col in range(M.ncols()):

f = monoms[col][:i] + monoms[col][i+1:]

if min([f[j]-e[j] for j in range(n)]) >= 0:

M[row,col] = prod([ binomial(f[j],e[j]) * pt[j]**(f[j]-e[j])

for j in (k for k in range(n) if f[k] > e[k]) ])

return M

def _morphism(self, *args, **kwds):

"""

Construct a morphism.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: P2._morphism(P2.Hom(P2), [x,y,z])

Scheme endomorphism of Projective Space of dimension 2 over Finite Field of size 3

Defn: Defined on coordinates by sending (x : y : z) to

(x : y : z)

"""

return SchemeMorphism_polynomial_projective_space(*args, **kwds)

def _point_homset(self, *args, **kwds):

"""

Construct a point Hom-set.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: P2._point_homset(Spec(GF(3)), P2)

Set of rational points of Projective Space of dimension 2 over Finite Field of size 3

"""

return SchemeHomset_points_projective_ring(*args, **kwds)

def point(self, v, check=True):

"""

Create a point on this projective space.

INPUT:

- ``v`` -- anything that defines a point

- ``check`` -- boolean (optional, default: ``True``); whether

to check the defining data for consistency

OUTPUT: A point of this projective space.

EXAMPLES::

sage: P2 = ProjectiveSpace(QQ, 2)

sage: P2.point([4,5])

(4 : 5 : 1)

sage: P = ProjectiveSpace(QQ, 1)

sage: P.point(infinity)

(1 : 0)

sage: P = ProjectiveSpace(QQ, 2)

sage: P.point(infinity)

Traceback (most recent call last):

...

ValueError: +Infinity not well defined in dimension > 1

sage: P = ProjectiveSpace(ZZ, 2)

sage: P.point([infinity])

Traceback (most recent call last):

...

ValueError: [+Infinity] not well defined in dimension > 1

"""

from sage.rings.infinity import infinity

if v is infinity or\

(isinstance(v, (list,tuple)) and len(v) == 1 and v[0] is infinity):

if self.dimension_relative() > 1:

raise ValueError("%s not well defined in dimension > 1"%v)

v = [1, 0]

return self.point_homset()(v, check=check)

def _point(self, *args, **kwds):

"""

Construct a point.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: point_homset = P2._point_homset(Spec(GF(3)), P2)

sage: P2._point(point_homset, [1,2,3])

(2 : 1 : 0)

"""

return SchemeMorphism_point_projective_ring(*args, **kwds)

def _repr_(self):

"""

Return a string representation of this projective space.

EXAMPLES::

sage: ProjectiveSpace(1, ZZ, 'x')

Projective Space of dimension 1 over Integer Ring

TESTS::

sage: ProjectiveSpace(3, Zp(5), 'y')._repr_()

'Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20'

"""

return "Projective Space of dimension %s over %s"%(self.dimension_relative(), self.base_ring())

def _repr_generic_point(self, v=None):

"""

Return a string representation of the generic point

corresponding to the list of polys ``v`` on this projective space.

If ``v`` is None, the representation of the generic point of

the projective space is returned.

EXAMPLES::

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)

sage: P._repr_generic_point([z*y-x^2])

'(-x^2 + y*z)'

sage: P._repr_generic_point()

'(x : y : z)'

"""

if v is None:

v = self.gens()

return '(%s)'%(" : ".join([repr(f) for f in v]))

def _latex_generic_point(self, v=None):

"""

Return a LaTeX representation of the generic point

corresponding to the list of polys ``v`` on this projective space.

If ``v`` is None, the representation of the generic point of

the projective space is returned.

EXAMPLES::

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)

sage: P._latex_generic_point([z*y-x^2])

'\\left(- x^{2} + y z\\right)'

sage: P._latex_generic_point()

'\\left(x : y : z\\right)'

"""

if v is None:

v = self.gens()

return '\\left(%s\\right)'%(" : ".join([str(latex(f)) for f in v]))

def change_ring(self, R):

r"""

Return a projective space over ring ``R``.

INPUT:

- ``R`` -- commutative ring or morphism.

OUTPUT:

- projective space over ``R``.

.. NOTE::

There is no need to have any relation between ``R`` and the base ring

of this space, if you want to have such a relation, use

``self.base_extend(R)`` instead.

EXAMPLES::

sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)

sage: PQ = P.change_ring(QQ); PQ

Projective Space of dimension 2 over Rational Field

sage: PQ.change_ring(GF(5))

Projective Space of dimension 2 over Finite Field of size 5

sage: K.<w> = QuadraticField(2)

sage: P = ProjectiveSpace(K,2,'t')

sage: P.change_ring(K.embeddings(QQbar)[0])

Projective Space of dimension 2 over Algebraic Field

"""

if isinstance(R, Map):

return ProjectiveSpace(self.dimension_relative(), R.codomain(),

self.variable_names())

else:

return ProjectiveSpace(self.dimension_relative(), R,

self.variable_names())

def is_projective(self):

"""

Return that this ambient space is projective `n`-space.

EXAMPLES::

sage: ProjectiveSpace(3,QQ).is_projective()

True

"""

return True

def subscheme(self, X):

"""

Return the closed subscheme defined by ``X``.

INPUT:

- ``X`` - a list or tuple of equations.

EXAMPLES::

sage: A.<x,y,z> = ProjectiveSpace(2, QQ)

sage: X = A.subscheme([x*z^2, y^2*z, x*y^2]); X

Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:

x*z^2,

y^2*z,

x*y^2

sage: X.defining_polynomials ()

(x*z^2, y^2*z, x*y^2)

sage: I = X.defining_ideal(); I

Ideal (x*z^2, y^2*z, x*y^2) of Multivariate Polynomial Ring in x, y, z over Rational Field

sage: I.groebner_basis()

[x*y^2, y^2*z, x*z^2]

sage: X.dimension()

sage: X.base_ring()

Rational Field

sage: X.base_scheme()

Spectrum of Rational Field

sage: X.structure_morphism()

Scheme morphism:

From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:

x*z^2,

y^2*z,

x*y^2

To: Spectrum of Rational Field

Defn: Structure map

sage: TestSuite(X).run(skip=["_test_an_element", "_test_elements",\

"_test_elements_eq", "_test_some_elements", "_test_elements_eq_reflexive",\

"_test_elements_eq_symmetric", "_test_elements_eq_transitive",\

"_test_elements_neq"])

"""

from sage.schemes.projective.projective_subscheme import AlgebraicScheme_subscheme_projective

return AlgebraicScheme_subscheme_projective(self, X)

def affine_patch(self, i, AA=None):

r"""

Return the `i^{th}` affine patch of this projective space.

This is an ambient affine space `\mathbb{A}^n_R,` where

`R` is the base ring of self, whose "projective embedding"

map is `1` in the `i^{th}` factor.

INPUT:

- ``i`` -- integer between 0 and dimension of self, inclusive.

- ``AA`` -- (default: None) ambient affine space, this is constructed

if it is not given.

OUTPUT:

- An ambient affine space with fixed projective_embedding map.

EXAMPLES::

sage: PP = ProjectiveSpace(5) / QQ

sage: AA = PP.affine_patch(2)

sage: AA

Affine Space of dimension 5 over Rational Field

sage: AA.projective_embedding()

Scheme morphism:

From: Affine Space of dimension 5 over Rational Field

To: Projective Space of dimension 5 over Rational Field

Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to

(x0 : x1 : 1 : x2 : x3 : x4)

sage: AA.projective_embedding(0)

Scheme morphism:

From: Affine Space of dimension 5 over Rational Field

To: Projective Space of dimension 5 over Rational Field

Defn: Defined on coordinates by sending (x0, x1, x2, x3, x4) to

(1 : x0 : x1 : x2 : x3 : x4)

sage: P.<x,y> = ProjectiveSpace(QQ,1)

sage: P.affine_patch(0).projective_embedding(0).codomain() == P

True

"""

i = int(i) # implicit type checking

n = self.dimension_relative()

if i < 0 or i > n:

raise ValueError("argument i (= %s) must be between 0 and %s"%(i, n))

try:

A = self.__affine_patches[i]

#assume that if you've passed in a new affine space you want to override

#the existing patch

if AA is None or A == AA:

return(A)

except AttributeError:

self.__affine_patches = {}

except KeyError:

pass

#if no ith patch exists, we may still be here with AA==None

if AA is None:

from sage.schemes.affine.affine_space import AffineSpace

AA = AffineSpace(n, self.base_ring(), names = 'x')

elif AA.dimension_relative() != n:

raise ValueError("affine space must be of the dimension %s"%(n))

AA._default_embedding_index = i

phi = AA.projective_embedding(i, self)

self.__affine_patches[i] = AA

return AA

def _an_element_(self):

r"""

Returns a (preferably typical) element of this space.

This is used both for illustration and testing purposes.

OUTPUT: a point in this projective space.

EXAMPLES::

sage: ProjectiveSpace(ZZ, 3, 'x').an_element()

(7 : 6 : 5 : 1)

sage: ProjectiveSpace(PolynomialRing(ZZ,'y'), 3, 'x').an_element()

(7*y : 6*y : 5*y : 1)

"""

n = self.dimension_relative()

R = self.base_ring()

return self([(7 - i) * R.an_element() for i in range(n)] + [R.one()])

def Lattes_map(self, E, m):

r"""

Given an elliptic curve ``E`` and an integer ``m`` return

the Lattes map associated to multiplication by `m`.

In other words, the rational map on the quotient

`E/\{\pm 1\} \cong \mathbb{P}^1` associated to `[m]:E \to E`.

INPUT:

- ``E`` -- an elliptic curve.

- ``m`` -- an integer.

OUTPUT: an endomorphism of this projective space.

Examples::

sage: P.<x,y> = ProjectiveSpace(QQ,1)

sage: E = EllipticCurve(QQ,[-1, 0])

sage: P.Lattes_map(E, 2)

Scheme endomorphism 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 + y^4 : 4*x^3*y - 4*x*y^3)

"""

if self.dimension_relative() != 1:

raise TypeError("must be dimension 1")

L = E.multiplication_by_m(m, x_only = True)

F = [L.numerator(), L.denominator()]

R = self.coordinate_ring()

x, y = R.gens()

phi = F[0].parent().hom([x],R)

F = [phi(F[0]).homogenize(y), phi(F[1]).homogenize(y)*y]

H = Hom(self,self)

return(H(F))

def cartesian_product(self, other):

r"""

Return the Cartesian product of this projective space and

``other``.

INPUT:

- ``other`` - A projective space with the same base ring as this space.

OUTPUT:

- A Cartesian product of projective spaces.

EXAMPLES::

sage: P1 = ProjectiveSpace(QQ, 1, 'x')

sage: P2 = ProjectiveSpace(QQ, 2, 'y')

sage: PP = P1.cartesian_product(P2); PP

Product of projective spaces P^1 x P^2 over Rational Field

sage: PP.gens()

(x0, x1, y0, y1, y2)

"""

from sage.schemes.product_projective.space import ProductProjectiveSpaces

return ProductProjectiveSpaces([self, other])

def chebyshev_polynomial(self, n, kind='first'):

"""

Generates an endomorphism of this projective line by a Chebyshev polynomial.

Chebyshev polynomials are a sequence of recursively defined orthogonal

polynomials. Chebyshev of the first kind are defined as `T_0(x) = 1`,

`T_1(x) = x`, and `T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)`. Chebyshev of

the second kind are defined as `U_0(x) = 1`,

`U_1(x) = 2x`, and `U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)`.

INPUT:

- ``n`` -- a non-negative integer.

- ``kind`` -- ``first`` or ``second`` specifying which kind of chebyshev the user would like

to generate. Defaults to ``first``.

OUTPUT: :class:`SchemeMorphism_polynomial_projective_space`

EXAMPLES::

sage: P.<x,y> = ProjectiveSpace(QQ, 1)

sage: P.chebyshev_polynomial(5, 'first')

Scheme endomorphism of Projective Space of dimension 1 over Rational Field

Defn: Defined on coordinates by sending (x : y) to

(16*x^5 - 20*x^3*y^2 + 5*x*y^4 : y^5)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)

sage: P.chebyshev_polynomial(3, 'second')

Scheme endomorphism of Projective Space of dimension 1 over Rational Field

Defn: Defined on coordinates by sending (x : y) to

(8*x^3 - 4*x*y^2 : y^3)

sage: P.<x,y> = ProjectiveSpace(QQ, 1)

sage: P.chebyshev_polynomial(3, 2)

Traceback (most recent call last):

...

ValueError: keyword 'kind' must have a value of either 'first' or 'second'

sage: P.<x,y> = ProjectiveSpace(QQ, 1)

sage: P.chebyshev_polynomial(-4, 'second')

Traceback (most recent call last):

...

ValueError: first parameter 'n' must be a non-negative integer

sage: P = ProjectiveSpace(QQ, 2, 'x')

sage: P.chebyshev_polynomial(2)

Traceback (most recent call last):

...

TypeError: projective space must be of dimension 1

"""

if self.dimension_relative() != 1:

raise TypeError("projective space must be of dimension 1")

n = ZZ(n)

if (n < 0):

raise ValueError("first parameter 'n' must be a non-negative integer")

#use the affine version and then homogenize.

A = self.affine_patch(1)

f = A.chebyshev_polynomial(n, kind)

return f.homogenize(1)

def veronese_embedding(self, d, CS=None, order='lex'):

r"""

Return the degree ``d`` Veronese embedding from this projective space.

INPUT:

- ``d`` -- a positive integer.

- ``CS`` -- a projective ambient space to embed into. If this projective space has dimension `N`, the

dimension of ``CS`` must be `\binom{N + d}{d} - 1`. This is constructed if not specified. Default:

``None``.

- ``order`` -- a monomial order to use to arrange the monomials defining the embedding. The monomials

will be arranged from greatest to least with respect to this order. Default: ``'lex'``.

OUTPUT:

- a scheme morphism from this projective space to ``CS``.

EXAMPLES::

sage: P.<x,y> = ProjectiveSpace(QQ, 1)

sage: vd = P.veronese_embedding(4, order='invlex')

sage: vd

Scheme morphism:

From: Projective Space of dimension 1 over Rational Field

To: Projective Space of dimension 4 over Rational Field

Defn: Defined on coordinates by sending (x : y) to

(y^4 : x*y^3 : x^2*y^2 : x^3*y : x^4)

Veronese surface::

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)

sage: Q.<q,r,s,t,u,v> = ProjectiveSpace(QQ, 5)

sage: vd = P.veronese_embedding(2, Q)

sage: vd

Scheme morphism:

From: Projective Space of dimension 2 over Rational Field

To: Projective Space of dimension 5 over Rational Field

Defn: Defined on coordinates by sending (x : y : z) to

(x^2 : x*y : x*z : y^2 : y*z : z^2)

sage: vd(P.subscheme([]))

Closed subscheme of Projective Space of dimension 5 over Rational Field

defined by:

-u^2 + t*v,

-s*u + r*v,

-s*t + r*u,

-s^2 + q*v,

-r*s + q*u,

-r^2 + q*t

"""

d = ZZ(d)

if d <= 0:

raise ValueError("(=%s) must be a positive integer"%d)

N = self.dimension()

# construct codomain space if not given

if CS is None:

CS = ProjectiveSpace(self.base_ring(), binomial(N + d, d) - 1)

else:

if not is_ProjectiveSpace(CS):

raise TypeError("(=%s) must be a projective space"%CS)

if CS.dimension() != binomial(N + d, d) - 1:

raise TypeError("(=%s) has the wrong dimension to serve as the codomain space"%CS)

R = self.coordinate_ring().change_ring(order=order)

monomials = sorted([R({tuple(v) : 1}) for v in WeightedIntegerVectors(d, [1] * (N + 1))])

monomials.reverse() # order the monomials greatest to least via the given monomial order

return Hom(self, CS)(monomials)

class ProjectiveSpace_field(ProjectiveSpace_ring):

def _point_homset(self, *args, **kwds):

"""

Construct a point Hom-set.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: P2._point_homset(Spec(GF(3)), P2)

Set of rational points of Projective Space of dimension 2 over Finite Field of size 3

"""

return SchemeHomset_points_projective_field(*args, **kwds)

def _point(self, *args, **kwds):

"""

Construct a point.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: point_homset = P2._point_homset(Spec(GF(3)), P2)

sage: P2._point(point_homset, [1,2,3])

(2 : 1 : 0)

"""

return SchemeMorphism_point_projective_field(*args, **kwds)

def _morphism(self, *args, **kwds):

"""

Construct a morphism.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: P2._morphism(P2.Hom(P2), [x,y,z])

Scheme endomorphism of Projective Space of dimension 2 over Finite Field of size 3

Defn: Defined on coordinates by sending (x : y : z) to

(x : y : z)

"""

return SchemeMorphism_polynomial_projective_space_field(*args, **kwds)

def points_of_bounded_height(self, bound, prec=53):

r"""

Returns an iterator of the points in self of absolute height of at most the given bound.

Bound check is strict for the rational field. Requires self to be projective space

over a number field. Uses the Doyle-Krumm algorithm for computing algebraic numbers

up to a given height [Doyle-Krumm]_.

INPUT:

- ``bound`` - a real number.

- ``prec`` - the precision to use to compute the elements of bounded height for number fields.

OUTPUT:

- an iterator of points in this space.

.. WARNING::

In the current implementation, the output of the [Doyle-Krumm]_ algorithm

cannot be guaranteed to be correct due to the necessity of floating point

computations. In some cases, the default 53-bit precision is

considerably lower than would be required for the algorithm to

generate correct output.

EXAMPLES::

sage: P.<x,y> = ProjectiveSpace(QQ, 1)

sage: list(P.points_of_bounded_height(5))

[(0 : 1), (1 : 1), (-1 : 1), (1/2 : 1), (-1/2 : 1), (2 : 1), (-2 : 1), (1/3 : 1),

(-1/3 : 1), (3 : 1), (-3 : 1), (2/3 : 1), (-2/3 : 1), (3/2 : 1), (-3/2 : 1), (1/4 : 1),

(-1/4 : 1), (4 : 1), (-4 : 1), (3/4 : 1), (-3/4 : 1), (4/3 : 1), (-4/3 : 1), (1 : 0)]

sage: u = QQ['u'].0

sage: P.<x,y,z> = ProjectiveSpace(NumberField(u^2 - 2, 'v'), 2)

sage: len(list(P.points_of_bounded_height(1.5)))

"""

if (is_RationalField(self.base_ring())):

ftype = False # stores whether the field is a number field or the rational field

elif (self.base_ring() in NumberFields()): # true for rational field as well, so check is_RationalField first

ftype = True

else:

raise NotImplementedError("self must be projective space over a number field")

bound = bound**(self.base_ring().absolute_degree()) # convert to relative height

n = self.dimension_relative()

R = self.base_ring()

zero = R(0)

i = n

while not i < 0:

P = [ zero for _ in range(i) ] + [ R(1) ] + [ zero for _ in range(n-i) ]

yield self(P)

if not ftype: # if rational field

iters = [ R.range_by_height(bound) for _ in range(i) ]

else: # if number field

iters = [ R.elements_of_bounded_height(bound, precision=prec) for _ in range(i) ]

for x in iters: next(x) # put at zero

j = 0

while j < i:

try:

P[j] = next(iters[j])

yield self(P)

j = 0

except StopIteration:

if not ftype: # if rational field

iters[j] = R.range_by_height(bound) # reset

else: # if number field

iters[j] = R.elements_of_bounded_height(bound, precision=prec) # reset

next(iters[j]) # put at zero

P[j] = zero

j += 1

i -= 1

def subscheme_from_Chow_form(self, Ch, dim):

r"""

Returns the subscheme defined by the Chow equations associated to the Chow form ``Ch``.

These equations define the subscheme set-theoretically, but only for smooth

subschemes and hypersurfaces do they define the subscheme as a scheme.

ALGORITHM:

The Chow form is a polynomial in the Plucker coordinates. The Plucker coordinates

are the bracket polynomials. We first re-write the Chow form in terms of the dual

Plucker coordinates. Then we expand `Ch(span(p,L)` for a generic point `p` and a

generic linear subspace `L`. The coefficients as polynomials in the coordinates

of `p` are the equations defining the subscheme. [DalbecSturmfels].

INPUT:

- ``Ch`` - a homogeneous polynomial.

- ``dim`` - the dimension of the associated scheme.

OUTPUT: a projective subscheme.

EXAMPLES::

sage: P = ProjectiveSpace(QQ, 4, 'z')

sage: R.<x0,x1,x2,x3,x4> = PolynomialRing(QQ)

sage: H = x1^2 + x2^2 + 5*x3*x4

sage: P.subscheme_from_Chow_form(H,3)

Closed subscheme of Projective Space of dimension 4 over Rational Field defined by:

-5*z0*z1 + z2^2 + z3^2

sage: P = ProjectiveSpace(QQ, 3, 'z')

sage: R.<x0,x1,x2,x3,x4,x5> = PolynomialRing(QQ)

sage: H = x1-x2-x3+x5+2*x0

sage: P.subscheme_from_Chow_form(H, 1)

Closed subscheme of Projective Space of dimension 3 over Rational Field

defined by:

-z1 + z3,

z0 + z2 + z3,

-z1 - 2*z3,

-z0 - z1 + 2*z2

sage: P.<x0,x1,x2,x3> = ProjectiveSpace(GF(7), 3)

sage: X = P.subscheme([x3^2+x1*x2,x2-x0])

sage: Ch = X.Chow_form();Ch

t0^2 - 2*t0*t3 + t3^2 - t2*t4 - t4*t5

sage: Y = P.subscheme_from_Chow_form(Ch, 1); Y

Closed subscheme of Projective Space of dimension 3 over Finite Field of

size 7 defined by:

x1*x2 + x3^2,

-x0*x2 + x2^2,

-x0*x1 - x1*x2 - 2*x3^2,

x0^2 - x0*x2,

x0*x1 + x3^2,

-2*x0*x3 + 2*x2*x3,

2*x0*x3 - 2*x2*x3,

x0^2 - 2*x0*x2 + x2^2

sage: I = Y.defining_ideal()

sage: I.saturation(I.ring().ideal(list(I.ring().gens())))[0]

Ideal (x0 - x2, x1*x2 + x3^2) of Multivariate Polynomial Ring in x0, x1,

x2, x3 over Finite Field of size 7

"""

if not Ch.is_homogeneous():

raise ValueError("Chow form must be a homogeneous polynomial")

n = self.dimension_relative()

R = Ch.parent()

if binomial(n+1,n-dim) != R.ngens():

raise ValueError("for given dimension, there should be %d variables in the Chow form" %binomial(n+1,n-dim))

vars = list(R.gens())

#create the brackets associated to variables

L1 = []

for t in UnorderedTuples(list(range(n + 1)), dim+1):

if all([t[i]<t[i+1] for i in range(dim)]):

L1.append(t)

#create the dual brackets

L2 = []

signs = []

for l in L1:

s = []

for v in range(n+1):

if not v in l:

s.append(v)

t1 = [b+1 for b in l]

t2 = [b+1 for b in s]

perm = Permutation(t1+t2)

signs.append(perm.sign())

L2.append(s)

#create the polys associated to dual brackets

if n-dim-1 > 0:

S = PolynomialRing(R.base_ring(),n+1,'z')

T = PolynomialRing(S,(n+1)*(n-dim-1),'s')

M = matrix(T,n-dim,n+1,list(S.gens())+list(T.gens()))

else:

T = PolynomialRing(R.base_ring(),n+1,'z')

M = matrix(T,n-dim,n+1,list(T.gens()))

coords=[]

for i in range(len(L2)):

coords.append(signs[i]*M.matrix_from_columns(L2[i]).det())

#substitute in dual brackets to chow form

phi = R.hom(coords,T)

ch = phi(Ch)

#coefficients are polys in zs which are the chow equations for the chow form

if n-dim-1 > 0:

X = self.subscheme(ch.coefficients())

else:

X = self.subscheme(ch)

return X

def point_transformation_matrix(self, points_source, points_target):

r"""

Returns a unique element of PGL that transforms one set of points to another.

Given a projective space of degree n and a set of n+2 source points and a set of n+2 target

points in the same projective space, such that no n+1 points of each set are linearly dependent

finds the unique element of PGL that translates the source points to the target points.

Warning :: will not work over precision fields

INPUT:

- ``points_source`` - points in source projective space.

- ``points_target`` - points in target projective space.

OUTPUT: Transformation matrix - element of PGL.

EXAMPLES::

sage: P1.<a,b,c>=ProjectiveSpace(QQ, 2)

sage: points_source=[P1([1,4,1]),P1([1,2,2]),P1([3,5,1]),P1([1,-1,1])]

sage: points_target=[P1([5,-2,7]),P1([3,-2,3]),P1([6,-5,9]), P1([3,6,7])]

sage: m = P1.point_transformation_matrix(points_source, points_target); m

[ -13/59 -128/59 -25/59]

[538/177 8/59 26/177]

[ -45/59 -196/59 1]

sage: [P1(list(m*vector(list(points_source[i])))) == points_target[i] for i in range(4)]

[True, True, True, True]

sage: P.<a,b> = ProjectiveSpace(GF(13),1)

sage: points_source = [P([-6,7]), P([1,4]), P([3,2])]

sage: points_target = [P([-1,2]), P([0,2]), P([-1,6])]

sage: P.point_transformation_matrix(points_source, points_target)

[10 4]

[10 1]

sage: P.<a,b> = ProjectiveSpace(QQ,1)

sage: points_source = [P([-6,-4]), P([1,4]), P([3,2])]

sage: points_target = [P([-1,2]), P([0,2]), P([-7,-3])]

sage: P.point_transformation_matrix(points_source, points_target)

Traceback (most recent call last):

...

ValueError: source points not independent

sage: P.<a,b> = ProjectiveSpace(QQ,1)

sage: points_source = [P([-6,-1]), P([1,4]), P([3,2])]

sage: points_target = [P([-1,2]), P([0,2]), P([-2,4])]

sage: P.point_transformation_matrix(points_source, points_target)

Traceback (most recent call last):

...

ValueError: target points not independent

sage: P.<a,b,c>=ProjectiveSpace(QQ, 2)

sage: points_source=[P([1,4,1]),P([2,-7,9]),P([3,5,1])]

sage: points_target=[P([5,-2,7]),P([3,-2,3]),P([6,-5,9]),P([6,-1,1])]

sage: P.point_transformation_matrix(points_source, points_target)

Traceback (most recent call last):

...

ValueError: incorrect number of points in source, need 4 points

sage: P.<a,b,c>=ProjectiveSpace(QQ, 2)

sage: points_source=[P([1,4,1]),P([2,-7,9]),P([3,5,1]),P([1,-1,1])]

sage: points_target=[P([5,-2,7]),P([3,-2,3]),P([6,-5,9]),P([6,-1,1]),P([7,8,-9])]

sage: P.point_transformation_matrix(points_source, points_target)

Traceback (most recent call last):

...

ValueError: incorrect number of points in target, need 4 points

sage: P.<a,b,c>=ProjectiveSpace(QQ, 2)

sage: P1.<x,y,z>=ProjectiveSpace(QQ, 2)

sage: points_source=[P([1,4,1]),P([2,-7,9]),P([3,5,1]),P1([1,-1,1])]

sage: points_target=[P([5,-2,7]),P([3,-2,3]),P([6,-5,9]),P([6,-1,1])]

sage: P.point_transformation_matrix(points_source, points_target)

Traceback (most recent call last):

...

ValueError: source points not in self

sage: P.<a,b,c>=ProjectiveSpace(QQ, 2)

sage: P1.<x,y,z>=ProjectiveSpace(QQ, 2)

sage: points_source=[P([1,4,1]),P([2,-7,9]),P([3,5,1]),P([1,-1,1])]

sage: points_target=[P([5,-2,7]),P([3,-2,3]),P([6,-5,9]),P1([6,-1,1])]

sage: P.point_transformation_matrix(points_source, points_target)

Traceback (most recent call last):

...

ValueError: target points not in self

"""

r = self.base_ring()

n = self.dimension_relative()

P = ProjectiveSpace(r, n**2+2*n,'p')

# makes sure there aren't to few or two many points

if len(points_source)!= n + 2:

raise ValueError ("incorrect number of points in source, need %d points"%(n+2))

if len(points_target)!= n + 2:

raise ValueError ("incorrect number of points in target, need %d points"%(n+2))

if any([x.codomain()!=self for x in points_source]):

raise ValueError ("source points not in self")

if any([x.codomain()!=self for x in points_target]):

raise ValueError ("target points not in self")

# putting points as the rows of the matrix

Ms = matrix(r, [list(s) for s in points_source])

if any([m == 0 for m in Ms.minors(n+1)]):

raise ValueError("source points not independent")

Mt = matrix(r, [list(t) for t in points_target])

if any([l == 0 for l in Mt.minors(n+1)]):

raise ValueError("target points not independent")

A = matrix(P.coordinate_ring(), n+1, n+1, P.gens())

#transpose to get image points and then get the list of image points with columns

funct = (A*Ms.transpose()).columns()

eq = []

for k in range(n+2):# n+2 num f point and n is size of pts

eq = eq+ [funct[k][i]*points_target[k][j] - funct[k][j]*points_target[k][i]\

for i in range(0,n+1) for j in range(i+1, n+1)]

v = P.subscheme(eq)

w = v.rational_points()

return matrix(r, n+1, n+1, list(w[0]))

def curve(self,F):

r"""

Return a curve defined by ``F`` in this projective space.

INPUT:

- ``F`` -- a polynomial, or a list or tuple of polynomials in

the coordinate ring of this projective space.

EXAMPLES::

sage: P.<x,y,z> = ProjectiveSpace(QQ, 2)

sage: P.curve([y^2 - x*z])

Projective Plane Curve over Rational Field defined by y^2 - x*z

"""

from sage.schemes.curves.constructor import Curve

return Curve(F, self)

class ProjectiveSpace_finite_field(ProjectiveSpace_field):

def _point(self, *args, **kwds):

"""

Construct a point.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: point_homset = P2._point_homset(Spec(GF(3)), P2)

sage: P2._point(point_homset, [1,2,3])

(2 : 1 : 0)

"""

return SchemeMorphism_point_projective_finite_field(*args, **kwds)

def _morphism(self, *args, **kwds):

"""

Construct a morphism.

For internal use only. See :mod:`morphism` for details.

TESTS::

sage: P2.<x,y,z> = ProjectiveSpace(2, GF(3))

sage: P2._morphism(P2.Hom(P2), [x,y,z])

Scheme endomorphism of Projective Space of dimension 2 over Finite Field of size 3

Defn: Defined on coordinates by sending (x : y : z) to

(x : y : z)

"""

return SchemeMorphism_polynomial_projective_space_finite_field(*args, **kwds)

def __iter__(self):

r"""

Return iterator over the elements of this projective space.

Note that iteration is over the decomposition

`\mathbb{P}^n = \mathbb{A}A^n \cup \mathbb{P}^n-1`, where

`\mathbb{A}A^n` is the `n`-th affine patch and

`\mathbb{P}^n-1` is the hyperplane at infinity

`x_n = 0`.

EXAMPLES::

sage: FF = FiniteField(3)

sage: PP = ProjectiveSpace(0,FF)

sage: [ x for x in PP ]

[(1)]

sage: PP = ProjectiveSpace(1,FF)

sage: [ x for x in PP ]

[(0 : 1), (1 : 1), (2 : 1), (1 : 0)]

sage: PP = ProjectiveSpace(2,FF)

sage: [ x for x in PP ]

[(0 : 0 : 1),

(1 : 0 : 1),

(2 : 0 : 1),

(0 : 1 : 1),

(1 : 1 : 1),

(2 : 1 : 1),

(0 : 2 : 1),

(1 : 2 : 1),

(2 : 2 : 1),

(0 : 1 : 0),

(1 : 1 : 0),

(2 : 1 : 0),

(1 : 0 : 0)]

AUTHORS:

- David Kohel

.. TODO::

Iteration for point sets over finite fields, and return of

iter of point set over base field. Note that the point set does not

know whether this is a projective space or subscheme.

"""

n = self.dimension_relative()

R = self.base_ring()

zero = R(0)

i = n

while not i < 0:

P = [ zero for _ in range(i) ] + [ R(1) ] + [ zero for _ in range(n-i) ]

yield self(P)

iters = [ iter(R) for _ in range(i) ]

for x in iters: next(x) # put at zero

j = 0

while j < i:

try:

P[j] = next(iters[j])

yield self(P)

j = 0

except StopIteration:

iters[j] = iter(R) # reset

next(iters[j]) # put at zero

P[j] = zero

j += 1

i -= 1

def rational_points(self, F=None):

"""

Return the list of ``F``-rational points on this projective space,

where ``F`` is a given finite field, or the base ring of this space.

EXAMPLES::

sage: P = ProjectiveSpace(1, GF(3))

sage: P.rational_points()

[(0 : 1), (1 : 1), (2 : 1), (1 : 0)]

sage: P.rational_points(GF(3^2, 'b'))

[(0 : 1), (b : 1), (b + 1 : 1), (2*b + 1 : 1), (2 : 1), (2*b : 1), (2*b + 2 : 1), (b + 2 : 1), (1 : 1), (1 : 0)]

"""

if F is None:

return [ P for P in self ]

elif not is_FiniteField(F):

raise TypeError("second argument (= %s) must be a finite field"%F)

return [ P for P in self.base_extend(F) ]

def rational_points_dictionary(self):

r"""

Return dictionary of points.

OUTPUT:

- dictionary

EXAMPLES::

sage: P1 = ProjectiveSpace(GF(7),1,'x')

sage: P1.rational_points_dictionary()

{(0 : 1): 0,

(1 : 0): 7,

(1 : 1): 1,

(2 : 1): 2,

(3 : 1): 3,

(4 : 1): 4,

(5 : 1): 5,

(6 : 1): 6}

"""

n = self.dimension_relative()

R = self.base_ring()

D={}

zero = R(0)

i = n

index=0

while not i < 0:

P = [ zero for _ in range(i) ] + [ R(1) ] + [ zero for _ in range(n-i) ]

D.update({self(P):index})

index+=1

iters = [ iter(R) for _ in range(i) ]

for x in iters: next(x) # put at zero

j = 0

while j < i:

try:

P[j] = next(iters[j])

D.update({self(P):index})

index+=1

j = 0

except StopIteration:

iters[j] = iter(R) # reset

next(iters[j]) # put at zero

P[j] = zero

j += 1

i -= 1

return(D)

class ProjectiveSpace_rational_field(ProjectiveSpace_field):

def rational_points(self, bound=0):

r"""

Returns the projective points `(x_0:\cdots:x_n)` over

`\QQ` with `|x_i| \leq` bound.

ALGORITHM:

The very simple algorithm works as follows: every point

`(x_0:\cdots:x_n)` in projective space has a unique

largest index `i` for which `x_i` is not

zero. The algorithm then iterates downward on this

index. We normalize by choosing `x_i` positive. Then,

the points `x_0,\ldots,x_{i-1}` are the points of

affine `i`-space that are relatively prime to

`x_i`. We access these by using the Tuples method.

INPUT:

- ``bound`` - integer.

EXAMPLES::

sage: PP = ProjectiveSpace(0, QQ)

sage: PP.rational_points(1)

[(1)]

sage: PP = ProjectiveSpace(1, QQ)

sage: PP.rational_points(2)

[(-2 : 1), (-1 : 1), (0 : 1), (1 : 1), (2 : 1), (-1/2 : 1), (1/2 : 1), (1 : 0)]

sage: PP = ProjectiveSpace(2, QQ)

sage: PP.rational_points(2)

[(-2 : -2 : 1), (-1 : -2 : 1), (0 : -2 : 1), (1 : -2 : 1), (2 : -2 : 1),

(-2 : -1 : 1), (-1 : -1 : 1), (0 : -1 : 1), (1 : -1 : 1), (2 : -1 : 1),

(-2 : 0 : 1), (-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1), (2 : 0 : 1), (-2 :

1 : 1), (-1 : 1 : 1), (0 : 1 : 1), (1 : 1 : 1), (2 : 1 : 1), (-2 : 2 :

1), (-1 : 2 : 1), (0 : 2 : 1), (1 : 2 : 1), (2 : 2 : 1), (-1/2 : -1 :

1), (1/2 : -1 : 1), (-1 : -1/2 : 1), (-1/2 : -1/2 : 1), (0 : -1/2 : 1),

(1/2 : -1/2 : 1), (1 : -1/2 : 1), (-1/2 : 0 : 1), (1/2 : 0 : 1), (-1 :

1/2 : 1), (-1/2 : 1/2 : 1), (0 : 1/2 : 1), (1/2 : 1/2 : 1), (1 : 1/2 :

1), (-1/2 : 1 : 1), (1/2 : 1 : 1), (-2 : 1 : 0), (-1 : 1 : 0), (0 : 1 :

0), (1 : 1 : 0), (2 : 1 : 0), (-1/2 : 1 : 0), (1/2 : 1 : 0), (1 : 0 :

0)]

AUTHORS:

- Benjamin Antieau (2008-01-12)

"""

if not bound > 0:

raise ValueError("argument bound (= %s) must be a positive integer")

n = self.dimension_relative()

Q = [k-bound for k in range(2*bound+1)] # the affine coordinates

R = [(k+1) for k in range(bound)] # the projective coordinate

S = [Tuples(Q, (k+1)) for k in range(n)]

pts = []

i = n

while i > 0:

P = [ 0 for _ in range(n+1) ]

for ai in R:

P[i] = ai

for tup in S[i-1]:

if gcd([ai] + tup) == 1:

for j in range(i):

P[j] = tup[j]

pts.append(self(P))

i -= 1

# now do i=0; this is treated as a special case so that

# we don't have all points (1:0),(2,0),(3,0),etc.

P = [ 0 for _ in range(n+1) ]; P[0] = 1

pts.append(self(P))

return pts

#fix the pickles from moving projective_space.py

from sage.structure.sage_object import register_unpickle_override

register_unpickle_override('sage.schemes.generic.projective_space',

'ProjectiveSpace_field',

ProjectiveSpace_field)

register_unpickle_override('sage.schemes.generic.projective_space',

'ProjectiveSpace_rational_field',

ProjectiveSpace_rational_field)

Coverage for local/lib/python2.7/site-packages/sage/schemes/projective/projective_space.py : 77%

437 statements 337 run 100 missing 0 excluded