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 > 0k
– an integer > 0m
– an integer (depending only on mod \(p^k\))zvec
,nzvec
– a list of integers inrange(Q.dim())
, orNone
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_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
– an integer \(\geq 1\)p
– a prime number > 2m
– an integerQdet
– a integer which is non-zero mod \(p\)
OUTPUT: an 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]