def count_modp_solutions__by_Gauss_sum(self, p, m):
"""
Return the number of solutions of `Q(x) = m (mod p)` of a
non-degenerate quadratic form over the finite field `Z/pZ`,
where `p` is a prime number > 2.
.. NOTE::
We adopt the useful convention that a zero-dimensional
quadratic form has exactly one solution always (i.e. the empty
vector).
These are defined in Table 1 on p363 of Hanke's "Local Densities..." paper.
INPUT:
- `p` -- a prime number > 2
- `m` -- an integer
OUTPUT: an integer >= 0
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: [Q.count_modp_solutions__by_Gauss_sum(3, m) for m in range(3)]
[9, 6, 12]
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,2])
sage: [Q.count_modp_solutions__by_Gauss_sum(3, m) for m in range(3)]
[9, 12, 6]
"""
if self.dim() == 0:
return 1
return count_modp__by_gauss_sum(self.dim(), p, m, self.Gram_det())
def count_modp_solutions__by_Gauss_sum(self, p, m):
"""
Returns the number of solutions of `Q(x) = m (mod p)` of a
non-degenerate quadratic form over the finite field `Z/pZ`,
where `p` is a prime number > 2.
Note: We adopt the useful convention that a zero-dimensional
quadratic form has exactly one solution always (i.e. the empty
vector).
These are defined in Table 1 on p363 of Hanke's "Local
Densities..." paper.
INPUT:
`p` -- a prime number > 2
`m` -- an integer
OUTPUT:
an integer >= 0
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: [Q.count_modp_solutions__by_Gauss_sum(3, m) for m in range(3)]
[9, 6, 12]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,2])
sage: [Q.count_modp_solutions__by_Gauss_sum(3, m) for m in range(3)]
[9, 12, 6]
"""
if self.dim() == 0:
return 1
else:
return count_modp__by_gauss_sum(self.dim(), p, m, self.Gram_det())