Optimized counting of congruence solutions

sage.quadratic_forms.count_local_2.CountAllLocalTypesNaive(Q, p, k, m, zvec, nzvec)[source]

This is an internal routine, which is called by sage.quadratic_forms.quadratic_form.QuadraticForm.count_congruence_solutions_by_type QuadraticForm.count_congruence_solutions_by_type(). See the documentation of that method for more details.

INPUT:

  • Q – quadratic form over \(\ZZ\)

  • p – prime number > 0

  • k – integer > 0

  • m – integer (depending only on mod \(p^k\))

  • zvec, nzvec – list of integers in range(Q.dim()), or None

OUTPUT:

a list of six integers \(\ge 0\) representing the solution types: [All, Good, Zero, Bad, BadI, BadII]

EXAMPLES:

sage: from sage.quadratic_forms.count_local_2 import CountAllLocalTypesNaive
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3])
sage: CountAllLocalTypesNaive(Q, 3, 1, 1, None, None)
[6, 6, 0, 0, 0, 0]
sage: CountAllLocalTypesNaive(Q, 3, 1, 2, None, None)
[6, 6, 0, 0, 0, 0]
sage: CountAllLocalTypesNaive(Q, 3, 1, 0, None, None)
[15, 12, 1, 2, 0, 2]

>>> from sage.all import *
>>> from sage.quadratic_forms.count_local_2 import CountAllLocalTypesNaive
>>> Q = DiagonalQuadraticForm(ZZ, [Integer(1),Integer(2),Integer(3)])
>>> CountAllLocalTypesNaive(Q, Integer(3), Integer(1), Integer(1), None, None)
[6, 6, 0, 0, 0, 0]
>>> CountAllLocalTypesNaive(Q, Integer(3), Integer(1), Integer(2), None, None)
[6, 6, 0, 0, 0, 0]
>>> CountAllLocalTypesNaive(Q, Integer(3), Integer(1), Integer(0), None, None)
[15, 12, 1, 2, 0, 2]
sage.quadratic_forms.count_local_2.count_all_local_good_types_normal_form(Q, p, k, m, zvec, nzvec)[source]

This is an internal routine, which is called by sage.quadratic_forms.quadratic_form.QuadraticForm.local_good_density_congruence_even QuadraticForm.local_good_density_congruence_even(). See the documentation of that method for more details.

INPUT:

  • Q – quadratic form over \(\ZZ\) in local normal form at p with no zero blocks mod \(p^k\)

  • p – prime number > 0

  • k – integer > 0

  • m – non-negative integer (depending only on mod \(p^k\))

  • zvec, nzvec – list of integers in range(Q.dim()), or None

OUTPUT:

a non-negative integer giving the number of solutions of Good type.

EXAMPLES:

sage: from sage.quadratic_forms.count_local_2 import count_all_local_good_types_normal_form
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3])
sage: Q_local_at2 = Q.local_normal_form(2)
sage: Q_local_at3 = Q.local_normal_form(3)
sage: count_all_local_good_types_normal_form(Q_local_at2, 2, 3, 3, None, None)
64
sage: count_all_local_good_types_normal_form(Q_local_at2, 2, 3, 3, [0], None)
32
sage: count_all_local_good_types_normal_form(Q_local_at3, 3, 2, 1, None, None)
54

>>> from sage.all import *
>>> from sage.quadratic_forms.count_local_2 import count_all_local_good_types_normal_form
>>> Q = DiagonalQuadraticForm(ZZ, [Integer(1),Integer(2),Integer(3)])
>>> Q_local_at2 = Q.local_normal_form(Integer(2))
>>> Q_local_at3 = Q.local_normal_form(Integer(3))
>>> count_all_local_good_types_normal_form(Q_local_at2, Integer(2), Integer(3), Integer(3), None, None)
64
>>> count_all_local_good_types_normal_form(Q_local_at2, Integer(2), Integer(3), Integer(3), [Integer(0)], None)
32
>>> count_all_local_good_types_normal_form(Q_local_at3, Integer(3), Integer(2), Integer(1), None, None)
54
sage.quadratic_forms.count_local_2.count_modp__by_gauss_sum(n, p, m, Qdet)[source]

Return the number of solutions of \(Q(x) = m\) over the finite field \(\ZZ/p\ZZ\), where \(p\) is a prime number > 2 and \(Q\) is a non-degenerate quadratic form of dimension \(n \geq 1\) and has Gram determinant Qdet.

REFERENCE:

These are defined in Table 1 on p363 of Hanke’s “Local Densities…” paper.

INPUT:

  • n – integer \(\geq 1\)

  • p – a prime number > 2

  • m – integer

  • Qdet – a integer which is nonzero mod \(p\)

OUTPUT: integer \(\geq 0\)

EXAMPLES:

sage: from sage.quadratic_forms.count_local_2 import count_modp__by_gauss_sum

sage: count_modp__by_gauss_sum(3, 3, 0, 1)    # for Q = x^2 + y^2 + z^2  => Gram Det = 1 (mod 3)
9
sage: count_modp__by_gauss_sum(3, 3, 1, 1)    # for Q = x^2 + y^2 + z^2  => Gram Det = 1 (mod 3)
6
sage: count_modp__by_gauss_sum(3, 3, 2, 1)    # for Q = x^2 + y^2 + z^2  => Gram Det = 1 (mod 3)
12

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: [Q.count_congruence_solutions(3, 1, m, None, None)
....:    == count_modp__by_gauss_sum(3, 3, m, 1)
....:  for m in range(3)]
[True, True, True]


sage: count_modp__by_gauss_sum(3, 3, 0, 2)    # for Q = x^2 + y^2 + 2*z^2  => Gram Det = 2 (mod 3)
9
sage: count_modp__by_gauss_sum(3, 3, 1, 2)    # for Q = x^2 + y^2 + 2*z^2  => Gram Det = 2 (mod 3)
12
sage: count_modp__by_gauss_sum(3, 3, 2, 2)    # for Q = x^2 + y^2 + 2*z^2  => Gram Det = 2 (mod 3)
6

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,2])
sage: [Q.count_congruence_solutions(3, 1, m, None, None)
....:    == count_modp__by_gauss_sum(3, 3, m, 2)
....:  for m in range(3)]
[True, True, True]

>>> from sage.all import *
>>> from sage.quadratic_forms.count_local_2 import count_modp__by_gauss_sum

>>> count_modp__by_gauss_sum(Integer(3), Integer(3), Integer(0), Integer(1))    # for Q = x^2 + y^2 + z^2  => Gram Det = 1 (mod 3)
9
>>> count_modp__by_gauss_sum(Integer(3), Integer(3), Integer(1), Integer(1))    # for Q = x^2 + y^2 + z^2  => Gram Det = 1 (mod 3)
6
>>> count_modp__by_gauss_sum(Integer(3), Integer(3), Integer(2), Integer(1))    # for Q = x^2 + y^2 + z^2  => Gram Det = 1 (mod 3)
12

>>> Q = DiagonalQuadraticForm(ZZ, [Integer(1),Integer(1),Integer(1)])
>>> [Q.count_congruence_solutions(Integer(3), Integer(1), m, None, None)
...    == count_modp__by_gauss_sum(Integer(3), Integer(3), m, Integer(1))
...  for m in range(Integer(3))]
[True, True, True]


>>> count_modp__by_gauss_sum(Integer(3), Integer(3), Integer(0), Integer(2))    # for Q = x^2 + y^2 + 2*z^2  => Gram Det = 2 (mod 3)
9
>>> count_modp__by_gauss_sum(Integer(3), Integer(3), Integer(1), Integer(2))    # for Q = x^2 + y^2 + 2*z^2  => Gram Det = 2 (mod 3)
12
>>> count_modp__by_gauss_sum(Integer(3), Integer(3), Integer(2), Integer(2))    # for Q = x^2 + y^2 + 2*z^2  => Gram Det = 2 (mod 3)
6

>>> Q = DiagonalQuadraticForm(ZZ, [Integer(1),Integer(1),Integer(2)])
>>> [Q.count_congruence_solutions(Integer(3), Integer(1), m, None, None)
...    == count_modp__by_gauss_sum(Integer(3), Integer(3), m, Integer(2))
...  for m in range(Integer(3))]
[True, True, True]