# 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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