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""" Counting Congruence Solutions
This file provides more user-friendly Python front-ends to the Cython code in :mod:`sage.quadratic_forms.count_local`. """ ################################################################## ## Methods for counting/computing the number of representations ## ## of a number by a quadratic form in Z/(p^k)Z of various types ## ##################################################################
from sage.quadratic_forms.count_local_2 import CountAllLocalTypesNaive
def count_congruence_solutions_as_vector(self, p, k, m, zvec, nzvec): """ Gives the number of integer solution vectors `x` satisfying the congruence Q(`x`) `= m (mod p^k)` of each solution type (i.e. All, Good, Zero, Bad, BadI, BadII) which satisfy the additional congruence conditions of having certain coefficients = 0 (mod `p`) and certain collections of coefficients not congruent to the zero vector (mod `p`).
A solution vector `x` satisfies the additional congruence conditions specified by zvec and nzvec (and therefore is counted) iff both of the following conditions hold:
1) `x[i] == 0 (mod p)` for all `i` in zvec
2) `x[i] != 0 (mod p) for all i` in nzvec
REFERENCES:
See Hanke's (????) paper "Local Densities and explicit bounds...", p??? for the definitions of the solution types and congruence conditions.
INPUT:
- `p` -- prime number > 0 - `k` -- an integer > 0 - `m` -- an integer (depending only on mod `p^k`) - zvec, nzvec -- a list of integers in range(self.dim()), or None
OUTPUT:
a list of six integers >= 0 representing the solution types: [All, Good, Zero, Bad, BadI, BadII]
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions_as_vector(3, 1, 1, [], []) [0, 0, 0, 0, 0, 0] sage: Q.count_congruence_solutions_as_vector(3, 1, 1, None, []) [0, 0, 0, 0, 0, 0] sage: Q.count_congruence_solutions_as_vector(3, 1, 1, [], None) [6, 6, 0, 0, 0, 0] sage: Q.count_congruence_solutions_as_vector(3, 1, 1, None, None) [6, 6, 0, 0, 0, 0] sage: Q.count_congruence_solutions_as_vector(3, 1, 2, None, None) [6, 6, 0, 0, 0, 0] sage: Q.count_congruence_solutions_as_vector(3, 1, 0, None, None) [15, 12, 1, 2, 0, 2]
"""
##/////////////////////////////////////////// ##/// Front-ends for our counting routines // ##///////////////////////////////////////////
def count_congruence_solutions(self, p, k, m, zvec, nzvec): """ Counts all solutions of Q(`x`) `= m (mod p^k)` satisfying the additional congruence conditions described in QuadraticForm.count_congruence_solutions_as_vector().
INPUT:
`p` -- prime number > 0
`k` -- an integer > 0
`m` -- an integer (depending only on mod `p^k`)
zvec, nzvec -- a list of integers in range(self.dim()), or None
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions(3, 1, 0, None, None) 15
"""
def count_congruence_solutions__good_type(self, p, k, m, zvec, nzvec): """ Counts the good-type solutions of Q(x) = m (mod p^k) satisfying the additional congruence conditions described in QuadraticForm.count_congruence_solutions_as_vector().
INPUT:
`p` -- prime number > 0
`k` -- an integer > 0
`m` -- an integer (depending only on mod `p^k`)
zvec, nzvec -- a list of integers up to dim(Q)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions__good_type(3, 1, 0, None, None) 12
"""
def count_congruence_solutions__zero_type(self, p, k, m, zvec, nzvec): """ Counts the zero-type solutions of Q(`x`) = `m (mod p^k)` satisfying the additional congruence conditions described in QuadraticForm.count_congruence_solutions_as_vector().
INPUT:
`p` -- prime number > 0
`k` -- an integer > 0
`m` -- an integer (depending only on mod `p^k`)
zvec, nzvec -- a list of integers up to dim(Q)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions__zero_type(3, 1, 0, None, None) 1
"""
def count_congruence_solutions__bad_type(self, p, k, m, zvec, nzvec): """ Counts the bad-type solutions of Q(`x`) `= m (mod p^k)` satisfying the additional congruence conditions described in QuadraticForm.count_congruence_solutions_as_vector().
INPUT:
`p` -- prime number > 0
`k` -- an integer > 0
`m` -- an integer (depending only on mod `p^k`)
zvec, nzvec -- a list of integers up to dim(Q)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions__bad_type(3, 1, 0, None, None) 2
"""
def count_congruence_solutions__bad_type_I(self, p, k, m, zvec, nzvec): """ Counts the bad-typeI solutions of Q(`x`) = `m (mod p^k)` satisfying the additional congruence conditions described in QuadraticForm.count_congruence_solutions_as_vector().
INPUT:
`p` -- prime number > 0
`k` -- an integer > 0
`m` -- an integer (depending only on mod `p^k`)
zvec, nzvec -- a list of integers up to dim(Q)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions__bad_type_I(3, 1, 0, None, None) 0
"""
def count_congruence_solutions__bad_type_II(self, p, k, m, zvec, nzvec): """ Counts the bad-typeII solutions of Q(`x`) `= m (mod p^k)` satisfying the additional congruence conditions described in QuadraticForm.count_congruence_solutions_as_vector().
INPUT:
`p` -- prime number > 0
`k` -- an integer > 0
`m` -- an integer (depending only on mod `p^k`)
zvec, nzvec -- a list of integers up to dim(Q)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3]) sage: Q.count_congruence_solutions__bad_type_II(3, 1, 0, None, None) 2
""" |