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r""" Enumeration of rational points on projective 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:
Projective, over `\QQ`::
sage: from sage.schemes.projective.projective_rational_point import enum_projective_rational_field sage: P.<X,Y,Z> = ProjectiveSpace(2,QQ) sage: C = P.subscheme([X+Y-Z]) sage: enum_projective_rational_field(C, 3) [(-2 : 3 : 1), (-1 : 1 : 0), (-1 : 2 : 1), (-1/2 : 3/2 : 1), (0 : 1 : 1), (1/3 : 2/3 : 1), (1/2 : 1/2 : 1), (2/3 : 1/3 : 1), (1 : 0 : 1), (3/2 : -1/2 : 1), (2 : -1 : 1), (3 : -2 : 1)]
Projective over a finite field::
sage: from sage.schemes.projective.projective_rational_point import enum_projective_finite_field sage: E = EllipticCurve('72').change_ring(GF(19)) sage: enum_projective_finite_field(E) [(0 : 1 : 0), (1 : 0 : 1), (3 : 0 : 1), (4 : 9 : 1), (4 : 10 : 1), (6 : 6 : 1), (6 : 13 : 1), (7 : 6 : 1), (7 : 13 : 1), (9 : 4 : 1), (9 : 15 : 1), (12 : 8 : 1), (12 : 11 : 1), (13 : 8 : 1), (13 : 11 : 1), (14 : 3 : 1), (14 : 16 : 1), (15 : 0 : 1), (16 : 9 : 1), (16 : 10 : 1), (17 : 7 : 1), (17 : 12 : 1), (18 : 9 : 1), (18 : 10 : 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. """
#***************************************************************************** # Copyright (C) 2010 William Stein, David Kohel, John Cremona, Charlie Turner # # 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/ #*****************************************************************************
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.projective.projective_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) """ raise TypeError("ambient space must be projective space over the rational field") else: raise TypeError("codomain must be projective space over the rational field")
""" Enumerates projective points on scheme ``X`` defined over a number field.
Simply checks all of the points of absolute height of at most ``B`` and adds those that are on the scheme to the list.
INPUT:
- ``X`` - a scheme defined over a number field.
- ``B`` - a real number.
- ``prec`` - the precision to use for computing the elements of bounded height of number fields.
OUTPUT:
- a list containing the projective points of ``X`` of absolute height up to ``B``, sorted.
.. WARNING::
In the current implementation, the output of the [Doyle-Krumm] algorithm for elements of bounded height 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: from sage.schemes.projective.projective_rational_point import enum_projective_number_field sage: u = QQ['u'].0 sage: K = NumberField(u^3 - 5,'v') sage: P.<x,y,z> = ProjectiveSpace(K, 2) sage: X = P.subscheme([x - y]) sage: enum_projective_number_field(X(K), 5^(1/3), prec=2^10) [(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1), (1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]
::
sage: u = QQ['u'].0 sage: K = NumberField(u^2 + 3, 'v') sage: A.<x,y> = ProjectiveSpace(K,1) sage: X = A.subscheme(x-y) sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field sage: enum_projective_number_field(X, 2) [(1 : 1)] """ raise TypeError("ambient space must be projective space over a number field") else: raise TypeError("codomain must be projective space over a number field")
""" 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.projective.projective_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). """ raise TypeError("ambient space must be projective space over a finite") else: raise TypeError("codomain must be projective space over a finite field")
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