Enumeration of rational points on affine schemes - Schemes

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# Enumeration of rational points on affine schemes

Naive algorithms for enumerating rational points over \(\QQ\) or finite fields over for general schemes.

Warning

Incorrect results and infinite loops may occur if using a wrong function.

(For instance using an affine function for a projective scheme or a finite field function for a scheme defined over an infinite field.)

EXAMPLES:

Affine, over \(\QQ\):

Sage

    sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
    sage: A.<x,y,z> = AffineSpace(3, QQ)
    sage: S = A.subscheme([2*x - 3*y])
    sage: enum_affine_rational_field(S, 2)
    [(0, 0, -2), (0, 0, -1), (0, 0, -1/2), (0, 0, 0),
     (0, 0, 1/2), (0, 0, 1), (0, 0, 2)]

Python

    >>> from sage.all import *
    >>> from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
    >>> A = AffineSpace(Integer(3), QQ, names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
    >>> S = A.subscheme([Integer(2)*x - Integer(3)*y])
    >>> enum_affine_rational_field(S, Integer(2))
    [(0, 0, -2), (0, 0, -1), (0, 0, -1/2), (0, 0, 0),
     (0, 0, 1/2), (0, 0, 1), (0, 0, 2)]

Affine over a finite field:

Sage

    sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
    sage: A.<w,x,y,z> = AffineSpace(4, GF(2))
    sage: enum_affine_finite_field(A(GF(2)))
    [(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 0),
     (0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 0, 0, 1),
     (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0),
     (1, 1, 1, 1)]

[Python]

    >>> from sage.all import *
    >>> from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
    >>> A = AffineSpace(Integer(4), GF(Integer(2)), names=('w', 'x', 'y', 'z',)); (w, x, y, z,) = A._first_ngens(4)
    >>> enum_affine_finite_field(A(GF(Integer(2))))
    [(0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 0),
     (0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 0, 0, 1),
     (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0),
     (1, 1, 1, 1)]

AUTHORS:

- David R. Kohel <kohel@maths.usyd.edu.au>: original version.

- John Cremona and Charlie Turner <charlotteturner@gmail.com> (06-2010): improvements to clarity and documentation.

sage.schemes.affine.affine_rational_point.enum_affine_finite_field(X)

  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

      sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
      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: 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)]

  Python

      >>> from sage.all import *
      >>> from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
      >>> F = GF(Integer(7))
      >>> A = AffineSpace(Integer(4), F, names=('w', 'x', 'y', 'z',)); (w, x, y, z,) = A._first_ngens(4)
      >>> C = A.subscheme([w**Integer(2) + x + Integer(4), y*z*x - Integer(6), z*y + w*x])
      >>> enum_affine_finite_field(C(F))
      []
      >>> C = A.subscheme([w**Integer(2) + x + Integer(4), y*z*x - Integer(6)])
      >>> 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

      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)]

  [Python]

      >>> from sage.all import *
      >>> A = AffineSpace(Integer(3), GF(Integer(3)), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> S = A.subscheme(x + y)
      >>> 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)

sage.schemes.affine.affine_rational_point.enum_affine_number_field(X, **kwds)

  Enumerates affine points on scheme `X` defined over a number field. Simply checks all of the points of absolute height up to `B` and adds those that are on the scheme to the list.

  This algorithm computes 2 lists: L containing elements x in \(K\) such that H_k(x) <= B, and a list L’ containing elements x in \(K\) that, due to floating point issues, may be slightly larger then the bound. This can be controlled by lowering the tolerance.

  ALGORITHM:

  This is an implementation of the revised algorithm (Algorithm 4) in [DK2013]. Algorithm 5 is used for imaginary quadratic fields.

  INPUT: keyword arguments:

  - `bound` – a real number

  - `tolerance` – a rational number in (0,1] used in Doyle-Krumm algorithm-4

  - `precision` – the precision to use for computing the elements of bounded height of number fields

  OUTPUT:

  - a list containing the affine points of `X` of absolute height up to `B`, sorted.

  EXAMPLES:

  Sage

      sage: # needs sage.rings.number_field
      sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
      sage: u = QQ['u'].0
      sage: K = NumberField(u^2 + 2, 'v')
      sage: A.<x,y,z> = AffineSpace(K, 3)
      sage: X = A.subscheme([y^2 - x])
      sage: enum_affine_number_field(X(K), bound=2**0.5)
      [(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v),
       (0, 0, v), (0, 0, 1), (1, -1, -1), (1, -1, -v), (1, -1, -1/2*v),
       (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1), (1, 1, -1),
       (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]

  Python

      >>> from sage.all import *
      >>> # needs sage.rings.number_field
      >>> from sage.schemes.affine.affine_rational_point import enum_affine_number_field
      >>> u = QQ['u'].gen(0)
      >>> K = NumberField(u**Integer(2) + Integer(2), 'v')
      >>> A = AffineSpace(K, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> X = A.subscheme([y**Integer(2) - x])
      >>> enum_affine_number_field(X(K), bound=Integer(2)**RealNumber('0.5'))
      [(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v),
       (0, 0, v), (0, 0, 1), (1, -1, -1), (1, -1, -v), (1, -1, -1/2*v),
       (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1), (1, 1, -1),
       (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]

  Sage

      sage: # needs sage.rings.number_field
      sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
      sage: u = QQ['u'].0
      sage: K = NumberField(u^2 + 3, 'v')
      sage: A.<x,y> = AffineSpace(K, 2)
      sage: X = A.subscheme(x - y)
      sage: enum_affine_number_field(X, bound=3**0.25)
      [(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2),
       (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2), (1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]

  [Python]

      >>> from sage.all import *
      >>> # needs sage.rings.number_field
      >>> from sage.schemes.affine.affine_rational_point import enum_affine_number_field
      >>> u = QQ['u'].gen(0)
      >>> K = NumberField(u**Integer(2) + Integer(3), 'v')
      >>> A = AffineSpace(K, Integer(2), names=('x', 'y',)); (x, y,) = A._first_ngens(2)
      >>> X = A.subscheme(x - y)
      >>> enum_affine_number_field(X, bound=Integer(3)**RealNumber('0.25'))
      [(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2),
       (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2), (1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]

sage.schemes.affine.affine_rational_point.enum_affine_rational_field(X, B)

  Enumerates affine rational points on scheme `X` 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

      sage: A.<x,y,z> = AffineSpace(3, QQ)
      sage: from sage.schemes.affine.affine_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)]

  Python

      >>> from sage.all import *
      >>> A = AffineSpace(Integer(3), QQ, names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3)
      >>> from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
      >>> enum_affine_rational_field(A(QQ), Integer(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

      sage: A.<w,x,y,z> = AffineSpace(4, QQ)
      sage: S = A.subscheme([x^2 - y*z + 1, w^3 + z + y^2])
      sage: enum_affine_rational_field(S(QQ), 1)
      [(0, 0, -1, -1)]
      sage: enum_affine_rational_field(S(QQ), 2)
      [(0, 0, -1, -1), (1, -1, -1, -2), (1, 1, -1, -2)]

  [Python]

      >>> from sage.all import *
      >>> A = AffineSpace(Integer(4), QQ, names=('w', 'x', 'y', 'z',)); (w, x, y, z,) = A._first_ngens(4)
      >>> S = A.subscheme([x**Integer(2) - y*z + Integer(1), w**Integer(3) + z + y**Integer(2)])
      >>> enum_affine_rational_field(S(QQ), Integer(1))
      [(0, 0, -1, -1)]
      >>> enum_affine_rational_field(S(QQ), Integer(2))
      [(0, 0, -1, -1), (1, -1, -1, -2), (1, 1, -1, -2)]

  Sage

      sage: A.<x,y> = AffineSpace(2, QQ)
      sage: C = Curve(x^2 + y - x)                                                    # needs sage.libs.singular
      sage: enum_affine_rational_field(C, 10)         # long time (3 s)               # needs sage.libs.singular
      [(-2, -6), (-1, -2), (-2/3, -10/9), (-1/2, -3/4), (-1/3, -4/9),
       (0, 0), (1/3, 2/9), (1/2, 1/4), (2/3, 2/9), (1, 0),
       (4/3, -4/9), (3/2, -3/4), (5/3, -10/9), (2, -2), (3, -6)]

  Python

      >>> from sage.all import *
      >>> A = AffineSpace(Integer(2), QQ, names=('x', 'y',)); (x, y,) = A._first_ngens(2)
      >>> C = Curve(x**Integer(2) + y - x)                                                    # needs sage.libs.singular
      >>> enum_affine_rational_field(C, Integer(10))         # long time (3 s)               # needs sage.libs.singular
      [(-2, -6), (-1, -2), (-2/3, -10/9), (-1/2, -3/4), (-1/3, -4/9),
       (0, 0), (1/3, 2/9), (1/2, 1/4), (2/3, 2/9), (1, 0),
       (4/3, -4/9), (3/2, -3/4), (5/3, -10/9), (2, -2), (3, -6)]

  AUTHORS:

  - David R. Kohel <kohel@maths.usyd.edu.au>: original version.

  - Charlie Turner (06-2010): small adjustments.

  - Raman Raghukul 2018: updated.