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:

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_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.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:

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