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: 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)]
Affine over a finite field:
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)]
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: 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.affine.affine_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)
-
sage.schemes.affine.affine_rational_point.
enum_affine_number_field
(X, B)¶ Enumerates affine points on scheme
X
defined over a number field. Simply checks all of the points of absolute height up toB
and adds those that are on the scheme to the list.INPUT:
X
- a scheme defined over a number field.B
- a real number.
OUTPUT:
- a list containing the affine points of
X
of absolute height up toB
, sorted.
EXAMPLES:
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), 4) [(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: 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: from sage.schemes.affine.affine_rational_point import enum_affine_number_field sage: enum_affine_number_field(X, 3) [(-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 boundB
.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 toB
, sorted.
EXAMPLES:
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)]
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 <kohel@maths.usyd.edu.au>: original version.
- Charlie Turner (06-2010): small adjustments.