Optimised Cython code for counting congruence solutions¶
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sage.quadratic_forms.count_local_2.
CountAllLocalTypesNaive
(Q, p, k, m, zvec, nzvec)¶ 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
- \(k\) – an integer > 0
- \(m\) – 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]
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sage.quadratic_forms.count_local_2.
count_modp__by_gauss_sum
(n, p, m, Qdet)¶ Returns the number of solutions of Q(x) = m over the finite field Z/pZ, where p is a prime number > 2 and Q is a non-degenerate quadratic form of dimension n >= 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 >= 1
- p – a prime number > 2
- m – an integer
- Qdet – a integer which is non-zero mod p
- OUTPUT:
- an integer >= 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]
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sage.quadratic_forms.count_local_2.
extract_sublist_indices
(Biglist, Smalllist)¶ Returns the indices of Biglist which index the entries of Smalllist appearing in Biglist. (Note that Smalllist may not be a sublist of Biglist.)
NOTE 1: This is an internal routine which deals with re-indexing lists, and is not exported to the QuadraticForm namespace!
NOTE 2: This should really by applied only when BigList has no repeated entries.
TO DO: * Please revisit this routine, and eliminate it! *
- INPUT:
- Biglist, Smalllist – two lists of a common type, where Biglist has no repeated entries.
- OUTPUT:
- a list of integers >= 0
EXAMPLES:
sage: from sage.quadratic_forms.count_local_2 import extract_sublist_indices sage: biglist = [1,3,5,7,8,2,4] sage: sublist = [5,3,2] sage: sublist == [biglist[i] for i in extract_sublist_indices(biglist, sublist)] ## Ok whenever Smalllist is a sublist of Biglist True sage: extract_sublist_indices([1,2,3,6,9,11], [1,3,2,9]) [0, 2, 1, 4] sage: extract_sublist_indices([1,2,3,6,9,11], [1,3,10,2,9,0]) [0, 2, 1, 4] sage: extract_sublist_indices([1,3,5,3,8], [1,5]) Traceback (most recent call last): ... TypeError: Biglist must not have repeated entries!