Coverage for local/lib/python2.7/site-packages/sage/quadratic_forms/quadratic_form__local_density_interfaces.py : 67%

""" 

Local Density Interfaces 

""" 

## // This is needed in the filter for primitivity... 

## #include "../max-min.h" 

from sage.arith.all import valuation 

from sage.rings.rational_field import QQ 

def local_density(self, p, m): 

""" 

Gives the local density -- should be called by the user. =) 

NOTE: This screens for imprimitive forms, and puts the quadratic 

form in local normal form, which is a *requirement* of the 

routines performing the computations! 

INPUT: 

`p` -- a prime number > 0 

`m` -- an integer 

OUTPUT: 

a rational number 

EXAMPLES:: 

sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) ## NOTE: This is already in local normal form for *all* primes p! 

sage: Q.local_density(p=2, m=1) 

1 

sage: Q.local_density(p=3, m=1) 

8/9 

sage: Q.local_density(p=5, m=1) 

24/25 

sage: Q.local_density(p=7, m=1) 

48/49 

sage: Q.local_density(p=11, m=1) 

120/121 

""" 

n = self.dim() 

if (n == 0): 

raise TypeError("Oops! We currently don't handle 0-dim'l forms. =(") 

## Find the local normal form and p-scale of Q -- Note: This uses the valuation ordering of local_normal_form. 

## TO DO: Write a separate p-scale and p-norm routines! 

Q_local = self.local_normal_form(p) 

if n == 1: 

p_valuation = valuation(Q_local[0,0], p) 

else: 

p_valuation = min(valuation(Q_local[0,0], p), valuation(Q_local[0,1], p)) 

## If m is less p-divisible than the matrix, return zero 

if ((m != 0) and (valuation(m,p) < p_valuation)): ## Note: The (m != 0) condition protects taking the valuation of zero. 

return QQ(0) 

## If the form is imprimitive, rescale it and call the local density routine 

p_adjustment = QQ(1) / p**p_valuation 

m_prim = QQ(m) / p**p_valuation 

Q_prim = Q_local.scale_by_factor(p_adjustment) 

## Return the densities for the reduced problem 

return Q_prim.local_density_congruence(p, m_prim) 

def local_primitive_density(self, p, m): 

""" 

Gives the local primitive density -- should be called by the user. =) 

NOTE: This screens for imprimitive forms, and puts the 

quadratic form in local normal form, which is a *requirement* of 

the routines performing the computations! 

INPUT: 

`p` -- a prime number > 0 

`m` -- an integer 

OUTPUT: 

a rational number 

EXAMPLES:: 

sage: Q = QuadraticForm(ZZ, 4, range(10)) 

sage: Q[0,0] = 5 

sage: Q[1,1] = 10 

sage: Q[2,2] = 15 

sage: Q[3,3] = 20 

sage: Q 

Quadratic form in 4 variables over Integer Ring with coefficients: 

[ 5 1 2 3 ] 

[ * 10 5 6 ] 

[ * * 15 8 ] 

[ * * * 20 ] 

sage: Q.theta_series(20) 

1 + 2*q^5 + 2*q^10 + 2*q^14 + 2*q^15 + 2*q^16 + 2*q^18 + O(q^20) 

sage: Q.local_normal_form(2) 

Quadratic form in 4 variables over Integer Ring with coefficients: 

[ 0 1 0 0 ] 

[ * 0 0 0 ] 

[ * * 0 1 ] 

[ * * * 0 ] 

sage: Q.local_primitive_density(2, 1) 

3/4 

sage: Q.local_primitive_density(5, 1) 

24/25 

sage: Q.local_primitive_density(2, 5) 

3/4 

sage: Q.local_density(2, 5) 

3/4 

""" 

n = self.dim() 

if (n == 0): 

raise TypeError("Oops! We currently don't handle 0-dim'l forms. =(") 

## Find the local normal form and p-scale of Q -- Note: This uses the valuation ordering of local_normal_form. 

## TO DO: Write a separate p-scale and p-norm routines! 

Q_local = self.local_normal_form(p) 

if n == 1: 

p_valuation = valuation(Q_local[0,0], p) 

else: 

p_valuation = min(valuation(Q_local[0,0], p), valuation(Q_local[0,1], p)) 

## If m is less p-divisible than the matrix, return zero 

if ((m != 0) and (valuation(m,p) < p_valuation)): ## Note: The (m != 0) condition protects taking the valuation of zero. 

return QQ(0) 

## If the form is imprimitive, rescale it and call the local density routine 

p_adjustment = QQ(1) / p**p_valuation 

m_prim = QQ(m) / p**p_valuation 

Q_prim = Q_local.scale_by_factor(p_adjustment) 

## Return the densities for the reduced problem 

return Q_prim.local_primitive_density_congruence(p, m_prim)