Enumeration of rational points on product projective schemes#

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

Warning

Incorrect results and infinite loops may occur if using a wrong function. (For instance using an affine function for a product projective scheme or a finite field function for a scheme defined over an infinite field.)

EXAMPLES:

Product Projective, over \(\QQ\):

sage: PP.<x,y,z> = ProductProjectiveSpaces([1,0], QQ)
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_rational_field
sage: enum_product_projective_rational_field(PP, 3)
[(-3 : 1 , 1), (-2 : 1 , 1), (-3/2 : 1 , 1),
 (-1 : 1 , 1), (-2/3 : 1 , 1), (-1/2 : 1 , 1),
 (-1/3 : 1 , 1), (0 : 1 , 1), (1/3 : 1 , 1),
 (1/2 : 1 , 1), (2/3 : 1 , 1), (1 : 0 , 1),
 (1 : 1 , 1), (3/2 : 1 , 1), (2 : 1 , 1),
 (3 : 1 , 1)]

Product projective over finite field:

sage: P1.<x,y,a,b> = ProductProjectiveSpaces([1, 1], GF(7))
sage: X = P1.subscheme([2*x + 3*y])
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_finite_field
sage: enum_product_projective_finite_field(X)
[(2 : 1 , 0 : 1), (2 : 1 , 1 : 0), (2 : 1 , 1 : 1),
 (2 : 1 , 2 : 1), (2 : 1 , 3 : 1), (2 : 1 , 4 : 1),
 (2 : 1 , 5 : 1), (2 : 1 , 6 : 1)]

AUTHORS:

Volker Braun and Ben Hutz (2014): initial version
Raghukul Raman (2018): code cleanup and added support for rational fields

sage.schemes.product_projective.rational_point.enum_product_projective_finite_field(X)#

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: PP.<x,y,z,w> = ProductProjectiveSpaces([1, 1], GF(3))
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_finite_field
sage: enum_product_projective_finite_field(PP)
[(0 : 1 , 0 : 1), (0 : 1 , 1 : 0), (0 : 1 , 1 : 1),
 (0 : 1 , 2 : 1), (1 : 0 , 0 : 1), (1 : 0 , 1 : 0),
 (1 : 0 , 1 : 1), (1 : 0 , 2 : 1), (1 : 1 , 0 : 1),
 (1 : 1 , 1 : 0), (1 : 1 , 1 : 1), (1 : 1 , 2 : 1),
 (2 : 1 , 0 : 1), (2 : 1 , 1 : 0), (2 : 1 , 1 : 1),
 (2 : 1 , 2 : 1)]

sage: PP.<x0,x1,x2,x3> = ProductProjectiveSpaces([1, 1], GF(17))
sage: X = PP.subscheme([x0^2 + 2*x1^2])
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_finite_field
sage: len(enum_product_projective_finite_field(X))
36

sage.schemes.product_projective.rational_point.enum_product_projective_number_field(X, **kwds)#

Enumerates product 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.

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:

kwds:

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 product projective points of X of absolute height up to B, sorted.

EXAMPLES:

sage: # needs sage.rings.number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 2, 'v')
sage: PP.<x,y,z,w> = ProductProjectiveSpaces([1, 1], K)
sage: X = PP.subscheme([x^2 + 2*y^2])
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_number_field
sage: enum_product_projective_number_field(X, bound=1.5)
[(-v : 1 , -1 : 1), (-v : 1 , -v : 1), (-v : 1 , -1/2*v : 1),
 (-v : 1 , 0 : 1), (-v : 1 , 1/2*v : 1), (-v : 1 , v : 1),
 (-v : 1 , 1 : 0), (-v : 1 , 1 : 1), (v : 1 , -1 : 1),
 (v : 1 , -v : 1), (v : 1 , -1/2*v : 1), (v : 1 , 0 : 1),
 (v : 1 , 1/2*v : 1), (v : 1 , v : 1), (v : 1 , 1 : 0),
 (v : 1 , 1 : 1)]

sage.schemes.product_projective.rational_point.enum_product_projective_rational_field(X, B)#

Enumerate 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 product projective points of X of height up to B, sorted.

EXAMPLES:

sage: PP.<x0,x1,x2,x3,x4> = ProductProjectiveSpaces([1, 2], QQ)
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_rational_field
sage: enum_product_projective_rational_field(PP, 1)
[(-1 : 1 , -1 : -1 : 1), (-1 : 1 , -1 : 0 : 1), (-1 : 1 , -1 : 1 : 0),
 (-1 : 1 , -1 : 1 : 1), (-1 : 1 , 0 : -1 : 1), (-1 : 1 , 0 : 0 : 1),
 (-1 : 1 , 0 : 1 : 0), (-1 : 1 , 0 : 1 : 1), (-1 : 1 , 1 : -1 : 1),
 (-1 : 1 , 1 : 0 : 0), (-1 : 1 , 1 : 0 : 1), (-1 : 1 , 1 : 1 : 0),
 (-1 : 1 , 1 : 1 : 1), (0 : 1 , -1 : -1 : 1), (0 : 1 , -1 : 0 : 1),
 (0 : 1 , -1 : 1 : 0), (0 : 1 , -1 : 1 : 1), (0 : 1 , 0 : -1 : 1),
 (0 : 1 , 0 : 0 : 1), (0 : 1 , 0 : 1 : 0), (0 : 1 , 0 : 1 : 1),
 (0 : 1 , 1 : -1 : 1), (0 : 1 , 1 : 0 : 0), (0 : 1 , 1 : 0 : 1),
 (0 : 1 , 1 : 1 : 0), (0 : 1 , 1 : 1 : 1), (1 : 0 , -1 : -1 : 1),
 (1 : 0 , -1 : 0 : 1), (1 : 0 , -1 : 1 : 0), (1 : 0 , -1 : 1 : 1),
 (1 : 0 , 0 : -1 : 1), (1 : 0 , 0 : 0 : 1), (1 : 0 , 0 : 1 : 0),
 (1 : 0 , 0 : 1 : 1), (1 : 0 , 1 : -1 : 1), (1 : 0 , 1 : 0 : 0),
 (1 : 0 , 1 : 0 : 1), (1 : 0 , 1 : 1 : 0), (1 : 0 , 1 : 1 : 1),
 (1 : 1 , -1 : -1 : 1), (1 : 1 , -1 : 0 : 1), (1 : 1 , -1 : 1 : 0),
 (1 : 1 , -1 : 1 : 1), (1 : 1 , 0 : -1 : 1), (1 : 1 , 0 : 0 : 1),
 (1 : 1 , 0 : 1 : 0), (1 : 1 , 0 : 1 : 1), (1 : 1 , 1 : -1 : 1),
 (1 : 1 , 1 : 0 : 0), (1 : 1 , 1 : 0 : 1), (1 : 1 , 1 : 1 : 0),
 (1 : 1 , 1 : 1 : 1)]

sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces([2, 1], QQ)
sage: X = PP.subscheme([x^2 + x*y + y*z, u*u - v*u])
sage: from sage.schemes.product_projective.rational_point import \
        enum_product_projective_rational_field
sage: enum_product_projective_rational_field(X, 4)
[(-2 : 4 : 1 , 0 : 1), (-2 : 4 : 1 , 1 : 1), (-1 : 1 : 0 , 0 : 1),
 (-1 : 1 : 0 , 1 : 1), (-2/3 : -4/3 : 1 , 0 : 1), (-2/3 : -4/3 : 1 , 1 : 1),
 (-1/2 : -1/2 : 1 , 0 : 1), (-1/2 : -1/2 : 1 , 1 : 1),
 (0 : 0 : 1 , 0 : 1), (0 : 0 : 1 , 1 : 1), (0 : 1 : 0 , 0 : 1),
 (0 : 1 : 0 , 1 : 1), (1 : -1/2 : 1 , 0 : 1), (1 : -1/2 : 1 , 1 : 1)]

sage.schemes.product_projective.rational_point.sieve(X, bound)#

Returns the list of all rational points on scheme X of height up to bound.

ALGORITHM:

Main idea behind the algorithm is to find points modulo primes and then reconstruct them using chinese remainder theorem. We compute the points modulo primes parallelly and then lift them via chinese remainder theorem in parallel. The LLL reduction algorithm is applied to each component of the points, and finally the result is merged and converted to a point on the subscheme.

For the algorithm to work correctly, sufficient primes need to be chosen, these are determined using the bounds dependent on the bound given in [Hutz2015].

INPUT:

X - a scheme with ambient space defined over a product of projective spaces
bound - a positive integer bound

OUTPUT:

a list containing the rational points of X of height up to bound, sorted

EXAMPLES:

sage: from sage.schemes.product_projective.rational_point import sieve
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces([2, 1], QQ)
sage: X = PP.subscheme([x^2 + y^2 - x*z, u*u - v*u])
sage: sieve(X, 2)                                                               # needs sage.libs.singular
[(0 : 0 : 1 , 0 : 1), (0 : 0 : 1 , 1 : 1), (1/2 : -1/2 : 1 , 0 : 1),
 (1/2 : -1/2 : 1 , 1 : 1), (1/2 : 1/2 : 1 , 0 : 1), (1/2 : 1/2 : 1 , 1 : 1),
 (1 : 0 : 1 , 0 : 1), (1 : 0 : 1 , 1 : 1)]