def Jacobian(X, **kwds):
"""
Return the Jacobian.
INPUT:
- ``X`` -- polynomial, algebraic variety, or anything else that
has a Jacobian elliptic curve.
- ``kwds`` -- optional keyword arguments.
The input ``X`` can be one of the following:
* A polynomial, see :func:`Jacobian_of_equation` for details.
* A curve, see :func:`Jacobian_of_curve` for details.
EXAMPLES::
sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(C, morphism=True)
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
u^3 + v^3 + w^3
To: Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
Defn: Defined on coordinates by sending (u : v : w) to
(u*v^7*w + u*v^4*w^4 + u*v*w^7 :
v^9 + 3/2*v^6*w^3 - 3/2*v^3*w^6 - w^9 :
-v^6*w^3 - v^3*w^6)
"""
try:
return X.jacobian(**kwds)
except AttributeError:
pass
morphism = kwds.pop('morphism', False)
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
if is_MPolynomial(X):
if morphism:
from sage.schemes.plane_curves.constructor import Curve
return Jacobian_of_equation(X, curve=Curve(X), **kwds)
else:
return Jacobian_of_equation(X, **kwds)
from sage.schemes.all import is_Scheme
if is_Scheme(X) and X.dimension() == 1:
return Jacobian_of_curve(X, morphism=morphism, **kwds)
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of Schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
\
sage: phi = ZZ.hom(QQ); phi
Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
"""
from sage.rings.all import is_CommutativeRing, is_RingHomomorphism
from sage.schemes.all import is_Scheme, Spec, is_SchemeMorphism
if is_Scheme(x) or is_SchemeMorphism(x):
return x
elif is_CommutativeRing(x):
return Spec(x)
elif is_RingHomomorphism(x):
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError, "No way to create an object or morphism in %s from %s"%(self, x)
def __call__(self, x):
"""
Call syntax for Spec.
INPUT/OUTPUT:
The argument ``x`` must be one of the following:
- a prime ideal of the coordinate ring; the output will
be the corresponding point of X
- an element (or list of elements) of the coordinate ring
which generates a prime ideal; the output will be the
corresponding point of X
- a ring or a scheme S; the output will be the set X(S) of
S-valued points on X
EXAMPLES::
sage: S = Spec(ZZ)
sage: P = S(3); P
Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring
sage: type(P)
<class 'sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal'>
sage: S(ZZ.ideal(next_prime(1000000)))
Point on Spectrum of Integer Ring defined by the Principal ideal (1000003) of Integer Ring
sage: R.<x, y, z> = QQ[]
sage: S = Spec(R)
sage: P = S(R.ideal(x, y, z)); P
Point on Spectrum of Multivariate Polynomial Ring
in x, y, z over Rational Field defined by the Ideal (x, y, z)
of Multivariate Polynomial Ring in x, y, z over Rational Field
This indicates the fix of :trac:`12734`::
sage: S = Spec(ZZ)
sage: S(ZZ)
Set of rational points of Spectrum of Integer Ring
sage: S(S)
Set of rational points of Spectrum of Integer Ring
"""
if is_CommutativeRing(x):
return self.point_homset(x)
from sage.schemes.all import is_Scheme
if is_Scheme(x):
return x.Hom(self)
return SchemeTopologicalPoint_prime_ideal(self, x)
def __call__(self, x):
"""
Call syntax for Spec.
INPUT/OUTPUT:
The argument ``x`` must be one of the following:
- a prime ideal of the coordinate ring; the output will
be the corresponding point of X
- an element (or list of elements) of the coordinate ring
which generates a prime ideal; the output will be the
corresponding point of X
- a ring or a scheme S; the output will be the set X(S) of
S-valued points on X
EXAMPLES::
sage: S = Spec(ZZ)
sage: P = S(3); P
Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring
sage: type(P)
<class 'sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal'>
sage: S(ZZ.ideal(next_prime(1000000)))
Point on Spectrum of Integer Ring defined by the Principal ideal (1000003) of Integer Ring
sage: R.<x, y, z> = QQ[]
sage: S = Spec(R)
sage: P = S(R.ideal(x, y, z)); P
Point on Spectrum of Multivariate Polynomial Ring
in x, y, z over Rational Field defined by the Ideal (x, y, z)
of Multivariate Polynomial Ring in x, y, z over Rational Field
This indicates the fix of :trac:`12734`::
sage: S = Spec(ZZ)
sage: S(ZZ)
Set of rational points of Spectrum of Integer Ring
sage: S(S)
Set of rational points of Spectrum of Integer Ring
"""
if is_CommutativeRing(x):
return self.point_homset(x)
from sage.schemes.all import is_Scheme
if is_Scheme(x):
return x.Hom(self)
return SchemeTopologicalPoint_prime_ideal(self, x)
def Schemes(X=None):
"""
Construct a category of schemes.
EXAMPLES::
sage: Schemes()
Category of Schemes
sage: Schemes(Spec(ZZ))
Category of schemes over Spectrum of Integer Ring
sage: Schemes(ZZ)
Category of schemes over Spectrum of Integer Ring
"""
if X is None:
return Schemes_abstract()
from sage.schemes.all import is_Scheme
if not is_Scheme(X):
X = Schemes()(X)
return Schemes_over_base(X)
def enum_projective_rational_field(X,B):
r"""
Enumerates projective, rational points on scheme ``X`` of height up to
bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the projective points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: P.<X,Y,Z> = ProjectiveSpace(2,QQ)
sage: C = P.subscheme([X+Y-Z])
sage: from sage.schemes.generic.rational_point import enum_projective_rational_field
sage: enum_projective_rational_field(C(QQ),6)
[(-5 : 6 : 1), (-4 : 5 : 1), (-3 : 4 : 1), (-2 : 3 : 1),
(-3/2 : 5/2 : 1), (-1 : 1 : 0), (-1 : 2 : 1), (-2/3 : 5/3 : 1),
(-1/2 : 3/2 : 1), (-1/3 : 4/3 : 1), (-1/4 : 5/4 : 1),
(-1/5 : 6/5 : 1), (0 : 1 : 1), (1/6 : 5/6 : 1), (1/5 : 4/5 : 1),
(1/4 : 3/4 : 1), (1/3 : 2/3 : 1), (2/5 : 3/5 : 1), (1/2 : 1/2 : 1),
(3/5 : 2/5 : 1), (2/3 : 1/3 : 1), (3/4 : 1/4 : 1), (4/5 : 1/5 : 1),
(5/6 : 1/6 : 1), (1 : 0 : 1), (6/5 : -1/5 : 1), (5/4 : -1/4 : 1),
(4/3 : -1/3 : 1), (3/2 : -1/2 : 1), (5/3 : -2/3 : 1), (2 : -1 : 1),
(5/2 : -3/2 : 1), (3 : -2 : 1), (4 : -3 : 1), (5 : -4 : 1),
(6 : -5 : 1)]
sage: enum_projective_rational_field(C,6) == enum_projective_rational_field(C(QQ),6)
True
::
sage: P3.<W,X,Y,Z> = ProjectiveSpace(3,QQ)
sage: enum_projective_rational_field(P3,1)
[(-1 : -1 : -1 : 1), (-1 : -1 : 0 : 1), (-1 : -1 : 1 : 0), (-1 : -1 : 1 : 1),
(-1 : 0 : -1 : 1), (-1 : 0 : 0 : 1), (-1 : 0 : 1 : 0), (-1 : 0 : 1 : 1),
(-1 : 1 : -1 : 1), (-1 : 1 : 0 : 0), (-1 : 1 : 0 : 1), (-1 : 1 : 1 : 0),
(-1 : 1 : 1 : 1), (0 : -1 : -1 : 1), (0 : -1 : 0 : 1), (0 : -1 : 1 : 0),
(0 : -1 : 1 : 1), (0 : 0 : -1 : 1), (0 : 0 : 0 : 1), (0 : 0 : 1 : 0),
(0 : 0 : 1 : 1), (0 : 1 : -1 : 1), (0 : 1 : 0 : 0), (0 : 1 : 0 : 1),
(0 : 1 : 1 : 0), (0 : 1 : 1 : 1), (1 : -1 : -1 : 1), (1 : -1 : 0 : 1),
(1 : -1 : 1 : 0), (1 : -1 : 1 : 1), (1 : 0 : -1 : 1), (1 : 0 : 0 : 0),
(1 : 0 : 0 : 1), (1 : 0 : 1 : 0), (1 : 0 : 1 : 1), (1 : 1 : -1 : 1),
(1 : 1 : 0 : 0), (1 : 1 : 0 : 1), (1 : 1 : 1 : 0), (1 : 1 : 1 : 1)]
ALGORITHM:
We just check all possible projective points in correct dimension
of projective space to see if they lie on ``X``.
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
zero = (0,) * n
pts = []
for c in cartesian_product_iterator([srange(-B,B+1) for _ in range(n)]):
if gcd(c) == 1 and c > zero:
try:
pts.append(X(c))
except TypeError:
pass
pts.sort()
return pts
def enum_affine_finite_field(X):
r"""
Enumerates affine points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the affine points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(7)
sage: A.<w,x,y,z> = AffineSpace(4,F)
sage: C = A.subscheme([w^2+x+4,y*z*x-6,z*y+w*x])
sage: from sage.schemes.generic.rational_point import enum_affine_finite_field
sage: enum_affine_finite_field(C(F))
[]
sage: C = A.subscheme([w^2+x+4,y*z*x-6])
sage: enum_affine_finite_field(C(F))
[(0, 3, 1, 2), (0, 3, 2, 1), (0, 3, 3, 3), (0, 3, 4, 4), (0, 3, 5, 6),
(0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
(1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
(2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
(3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
(4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
(5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
(5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
(6, 2, 5, 2), (6, 2, 6, 4)]
::
sage: A.<x,y,z> = AffineSpace(3,GF(3))
sage: S = A.subscheme(x+y)
sage: enum_affine_finite_field(S)
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2),
(2, 1, 0), (2, 1, 1), (2, 1, 2)]
ALGORITHM:
Checks all points in affine space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
F = X.value_ring()
pts = []
for c in cartesian_product_iterator([F]*n):
try:
pts.append(X(c))
except StandardError:
pass
pts.sort()
return pts
def enum_projective_finite_field(X):
"""
Enumerates projective points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the projective points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(53)
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: from sage.schemes.generic.rational_point import enum_projective_finite_field
sage: len(enum_projective_finite_field(P(F)))
2863
sage: 53^2+53+1
2863
::
sage: F = GF(9,'a')
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: C = Curve(X^3-Y^3+Z^2*Y)
sage: enum_projective_finite_field(C(F))
[(0 : 0 : 1), (0 : 1 : 1), (0 : 2 : 1), (1 : 1 : 0), (a + 1 : 2*a : 1),
(a + 1 : 2*a + 1 : 1), (a + 1 : 2*a + 2 : 1), (2*a + 2 : a : 1),
(2*a + 2 : a + 1 : 1), (2*a + 2 : a + 2 : 1)]
::
sage: F = GF(5)
sage: P2F.<X,Y,Z> = ProjectiveSpace(2,F)
sage: enum_projective_finite_field(P2F)
[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 2 : 1), (0 : 3 : 1), (0 : 4 : 1),
(1 : 0 : 0), (1 : 0 : 1), (1 : 1 : 0), (1 : 1 : 1), (1 : 2 : 1), (1 : 3 : 1),
(1 : 4 : 1), (2 : 0 : 1), (2 : 1 : 0), (2 : 1 : 1), (2 : 2 : 1), (2 : 3 : 1),
(2 : 4 : 1), (3 : 0 : 1), (3 : 1 : 0), (3 : 1 : 1), (3 : 2 : 1), (3 : 3 : 1),
(3 : 4 : 1), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1),
(4 : 4 : 1)]
ALGORITHM:
Checks all points in projective space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010).
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()-1
F = X.value_ring()
pts = []
for k in range(n+1):
for c in cartesian_product_iterator([F for _ in range(k)]):
try:
pts.append(X(list(c)+[1]+[0]*(n-k)))
except TypeError:
pass
pts.sort()
return pts
def enum_affine_rational_field(X,B):
"""
Enumerates affine rational points on scheme ``X`` (defined over `\QQ`) up
to bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the affine points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(3,QQ)
sage: from sage.schemes.generic.rational_point import enum_affine_rational_field
sage: enum_affine_rational_field(A(QQ),1)
[(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]
::
sage: A.<w,x,y,z> = AffineSpace(4,QQ)
sage: S = A.subscheme([x^2-y*z+3,w^3+z+y^2])
sage: enum_affine_rational_field(S(QQ),2)
[]
sage: enum_affine_rational_field(S(QQ),3)
[(-2, 0, -3, -1)]
::
sage: A.<x,y> = AffineSpace(2,QQ)
sage: C = Curve(x^2+y-x)
sage: enum_affine_rational_field(C,10)
[(-2, -6), (-1, -2), (0, 0), (1, 0), (2, -2), (3, -6)]
AUTHORS:
- David R. Kohel <*****@*****.**>: original version.
- Charlie Turner (06-2010): small adjustments.
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
if X.value_ring() is ZZ:
Q = [ 1 ]
else: # rational field
Q = range(1, B + 1)
R = [ 0 ] + [ s*k for k in range(1, B+1) for s in [1, -1] ]
pts = []
P = [0] * n
m = ZZ(0)
try:
pts.append(X(P))
except TypeError:
pass
iters = [ iter(R) for _ in range(n) ]
for it in iters:
it.next()
i = 0
while i < n:
try:
a = ZZ(iters[i].next())
except StopIteration:
iters[i] = iter(R) # reset
P[i] = iters[i].next() # reset P[i] to 0 and increment
i += 1
continue
m = m.gcd(a)
P[i] = a
for b in Q:
if m.gcd(b) == 1:
try:
pts.append(X([ num/b for num in P ]))
except TypeError:
pass
i = 0
m = ZZ(0)
pts.sort()
return pts
def enum_projective_rational_field(X,B):
r"""
Enumerates projective, rational points on scheme ``X`` of height up to
bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the projective points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: P.<X,Y,Z> = ProjectiveSpace(2,QQ)
sage: C = P.subscheme([X+Y-Z])
sage: from sage.schemes.generic.rational_point import enum_projective_rational_field
sage: enum_projective_rational_field(C(QQ),6)
[(-5 : 6 : 1), (-4 : 5 : 1), (-3 : 4 : 1), (-2 : 3 : 1),
(-3/2 : 5/2 : 1), (-1 : 1 : 0), (-1 : 2 : 1), (-2/3 : 5/3 : 1),
(-1/2 : 3/2 : 1), (-1/3 : 4/3 : 1), (-1/4 : 5/4 : 1),
(-1/5 : 6/5 : 1), (0 : 1 : 1), (1/6 : 5/6 : 1), (1/5 : 4/5 : 1),
(1/4 : 3/4 : 1), (1/3 : 2/3 : 1), (2/5 : 3/5 : 1), (1/2 : 1/2 : 1),
(3/5 : 2/5 : 1), (2/3 : 1/3 : 1), (3/4 : 1/4 : 1), (4/5 : 1/5 : 1),
(5/6 : 1/6 : 1), (1 : 0 : 1), (6/5 : -1/5 : 1), (5/4 : -1/4 : 1),
(4/3 : -1/3 : 1), (3/2 : -1/2 : 1), (5/3 : -2/3 : 1), (2 : -1 : 1),
(5/2 : -3/2 : 1), (3 : -2 : 1), (4 : -3 : 1), (5 : -4 : 1),
(6 : -5 : 1)]
sage: enum_projective_rational_field(C,6) == enum_projective_rational_field(C(QQ),6)
True
::
sage: P3.<W,X,Y,Z> = ProjectiveSpace(3,QQ)
sage: enum_projective_rational_field(P3,1)
[(-1 : -1 : -1 : 1), (-1 : -1 : 0 : 1), (-1 : -1 : 1 : 0), (-1 : -1 : 1 : 1),
(-1 : 0 : -1 : 1), (-1 : 0 : 0 : 1), (-1 : 0 : 1 : 0), (-1 : 0 : 1 : 1),
(-1 : 1 : -1 : 1), (-1 : 1 : 0 : 0), (-1 : 1 : 0 : 1), (-1 : 1 : 1 : 0),
(-1 : 1 : 1 : 1), (0 : -1 : -1 : 1), (0 : -1 : 0 : 1), (0 : -1 : 1 : 0),
(0 : -1 : 1 : 1), (0 : 0 : -1 : 1), (0 : 0 : 0 : 1), (0 : 0 : 1 : 0),
(0 : 0 : 1 : 1), (0 : 1 : -1 : 1), (0 : 1 : 0 : 0), (0 : 1 : 0 : 1),
(0 : 1 : 1 : 0), (0 : 1 : 1 : 1), (1 : -1 : -1 : 1), (1 : -1 : 0 : 1),
(1 : -1 : 1 : 0), (1 : -1 : 1 : 1), (1 : 0 : -1 : 1), (1 : 0 : 0 : 0),
(1 : 0 : 0 : 1), (1 : 0 : 1 : 0), (1 : 0 : 1 : 1), (1 : 1 : -1 : 1),
(1 : 1 : 0 : 0), (1 : 1 : 0 : 1), (1 : 1 : 1 : 0), (1 : 1 : 1 : 1)]
ALGORITHM:
We just check all possible projective points in correct dimension
of projective space to see if they lie on ``X``.
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
zero = (0,) * n
pts = []
for c in cartesian_product_iterator([srange(-B,B+1) for _ in range(n)]):
if gcd(c) == 1 and c > zero:
try:
pts.append(X(c))
except TypeError:
pass
pts.sort()
return pts
def enum_affine_finite_field(X):
r"""
Enumerates affine points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the affine points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(7)
sage: A.<w,x,y,z> = AffineSpace(4,F)
sage: C = A.subscheme([w^2+x+4,y*z*x-6,z*y+w*x])
sage: from sage.schemes.generic.rational_point import enum_affine_finite_field
sage: enum_affine_finite_field(C(F))
[]
sage: C = A.subscheme([w^2+x+4,y*z*x-6])
sage: enum_affine_finite_field(C(F))
[(0, 3, 1, 2), (0, 3, 2, 1), (0, 3, 3, 3), (0, 3, 4, 4), (0, 3, 5, 6),
(0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
(1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
(2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
(3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
(4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
(5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
(5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
(6, 2, 5, 2), (6, 2, 6, 4)]
::
sage: A.<x,y,z> = AffineSpace(3,GF(3))
sage: S = A.subscheme(x+y)
sage: enum_affine_finite_field(S)
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2),
(2, 1, 0), (2, 1, 1), (2, 1, 2)]
ALGORITHM:
Checks all points in affine space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
F = X.value_ring()
pts = []
for c in cartesian_product_iterator([F]*n):
try:
pts.append(X(c))
except StandardError:
pass
pts.sort()
return pts
def enum_projective_finite_field(X):
"""
Enumerates projective points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the projective points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(53)
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: from sage.schemes.generic.rational_point import enum_projective_finite_field
sage: len(enum_projective_finite_field(P(F)))
2863
sage: 53^2+53+1
2863
::
sage: F = GF(9,'a')
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: C = Curve(X^3-Y^3+Z^2*Y)
sage: enum_projective_finite_field(C(F))
[(0 : 0 : 1), (0 : 1 : 1), (0 : 2 : 1), (1 : 1 : 0), (a + 1 : 2*a : 1),
(a + 1 : 2*a + 1 : 1), (a + 1 : 2*a + 2 : 1), (2*a + 2 : a : 1),
(2*a + 2 : a + 1 : 1), (2*a + 2 : a + 2 : 1)]
::
sage: F = GF(5)
sage: P2F.<X,Y,Z> = ProjectiveSpace(2,F)
sage: enum_projective_finite_field(P2F)
[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 2 : 1), (0 : 3 : 1), (0 : 4 : 1),
(1 : 0 : 0), (1 : 0 : 1), (1 : 1 : 0), (1 : 1 : 1), (1 : 2 : 1), (1 : 3 : 1),
(1 : 4 : 1), (2 : 0 : 1), (2 : 1 : 0), (2 : 1 : 1), (2 : 2 : 1), (2 : 3 : 1),
(2 : 4 : 1), (3 : 0 : 1), (3 : 1 : 0), (3 : 1 : 1), (3 : 2 : 1), (3 : 3 : 1),
(3 : 4 : 1), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1),
(4 : 4 : 1)]
ALGORITHM:
Checks all points in projective space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010).
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()-1
F = X.value_ring()
pts = []
for k in range(n+1):
for c in cartesian_product_iterator([F for _ in range(k)]):
try:
pts.append(X(list(c)+[1]+[0]*(n-k)))
except TypeError:
pass
pts.sort()
return pts
def enum_affine_rational_field(X,B):
"""
Enumerates affine rational points on scheme ``X`` (defined over `\QQ`) up
to bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the affine points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(3,QQ)
sage: from sage.schemes.generic.rational_point import enum_affine_rational_field
sage: enum_affine_rational_field(A(QQ),1)
[(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]
::
sage: A.<w,x,y,z> = AffineSpace(4,QQ)
sage: S = A.subscheme([x^2-y*z+3,w^3+z+y^2])
sage: enum_affine_rational_field(S(QQ),2)
[]
sage: enum_affine_rational_field(S(QQ),3)
[(-2, 0, -3, -1)]
::
sage: A.<x,y> = AffineSpace(2,QQ)
sage: C = Curve(x^2+y-x)
sage: enum_affine_rational_field(C,10)
[(-2, -6), (-1, -2), (0, 0), (1, 0), (2, -2), (3, -6)]
AUTHORS:
- David R. Kohel <*****@*****.**>: original version.
- Charlie Turner (06-2010): small adjustments.
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
if X.value_ring() is ZZ:
Q = [ 1 ]
else: # rational field
Q = range(1, B + 1)
R = [ 0 ] + [ s*k for k in range(1, B+1) for s in [1, -1] ]
pts = []
P = [0] * n
m = ZZ(0)
try:
pts.append(X(P))
except TypeError:
pass
iters = [ iter(R) for _ in range(n) ]
for it in iters:
it.next()
i = 0
while i < n:
try:
a = ZZ(iters[i].next())
except StopIteration:
iters[i] = iter(R) # reset
P[i] = iters[i].next() # reset P[i] to 0 and increment
i += 1
continue
m = m.gcd(a)
P[i] = a
for b in Q:
if m.gcd(b) == 1:
try:
pts.append(X([ num/b for num in P ]))
except TypeError:
pass
i = 0
m = ZZ(0)
pts.sort()
return pts