def compute_definiteness_string_by_determinants(self):
"""
Compute the (positive) definiteness of a quadratic form by looking
at the signs of all of its upper-left subdeterminants. See also
self.compute_definiteness() for more documentation.
INPUT:
None
OUTPUT:
string describing the definiteness
EXAMPLES:
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1,1])
sage: Q.compute_definiteness_string_by_determinants()
'pos_def'
::
sage: Q = DiagonalQuadraticForm(ZZ, [])
sage: Q.compute_definiteness_string_by_determinants()
'zero'
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,0,-1])
sage: Q.compute_definiteness_string_by_determinants()
'degenerate'
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,-1])
sage: Q.compute_definiteness_string_by_determinants()
'indefinite'
::
sage: Q = DiagonalQuadraticForm(ZZ, [-1,-1])
sage: Q.compute_definiteness_string_by_determinants()
'neg_def'
"""
## Sanity Check
if not ((self.base_ring() == ZZ) or (self.base_ring() == QQ) or
(self.base_ring() == RR)):
raise NotImplementedError(
"Oops! We can only check definiteness over ZZ, QQ, and RR for now."
)
## Some useful variables
n = self.dim()
M = self.matrix()
## Deal with the zero-diml form
if n == 0:
return "zero"
## Deal with degenerate forms
if self.det() == 0:
return "degenerate"
## Check the sign of the ratios of consecutive determinants of the upper triangular r x r submatrices
first_coeff = self[0, 0]
for r in range(1, n + 1):
I = list(range(r))
new_det = M.matrix_from_rows_and_columns(I, I).det()
## Check for a (non-degenerate) zero -- so it's indefinite
if new_det == 0:
return "indefinite"
## Check for a change of signs in the upper r x r submatrix -- so it's indefinite
if sgn(first_coeff)**r != sgn(new_det):
return "indefinite"
## Here all ratios of determinants have the correct sign, so the matrix is (pos or neg) definite.
if first_coeff > 0:
return "pos_def"
else:
return "neg_def"
def compute_definiteness_string_by_determinants(self):
"""
Compute the (positive) definiteness of a quadratic form by looking
at the signs of all of its upper-left subdeterminants. See also
self.compute_definiteness() for more documentation.
INPUT:
None
OUTPUT:
string describing the definiteness
EXAMPLES:
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1,1])
sage: Q.compute_definiteness_string_by_determinants()
'pos_def'
::
sage: Q = DiagonalQuadraticForm(ZZ, [])
sage: Q.compute_definiteness_string_by_determinants()
'zero'
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,0,-1])
sage: Q.compute_definiteness_string_by_determinants()
'degenerate'
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,-1])
sage: Q.compute_definiteness_string_by_determinants()
'indefinite'
::
sage: Q = DiagonalQuadraticForm(ZZ, [-1,-1])
sage: Q.compute_definiteness_string_by_determinants()
'neg_def'
"""
## Sanity Check
if not ((self.base_ring() == ZZ) or (self.base_ring() == QQ) or (self.base_ring() == RR)):
raise NotImplementedError, "Oops! We can only check definiteness over ZZ, QQ, and RR for now."
## Some useful variables
n = self.dim()
M = self.matrix()
## Deal with the zero-diml form
if n == 0:
return "zero"
## Deal with degenerate forms
if self.det() == 0:
return "degenerate"
## Check the sign of the ratios of consecutive determinants of the upper triangular r x r submatrices
first_coeff = self[0,0]
for r in range(1,n+1):
I = range(r)
new_det = M.matrix_from_rows_and_columns(I, I).det()
## Check for a (non-degenerate) zero -- so it's indefinite
if new_det == 0:
return "indefinite"
## Check for a change of signs in the upper r x r submatrix -- so it's indefinite
if sgn(first_coeff)**r != sgn(new_det):
return "indefinite"
## Here all ratios of determinants have the correct sign, so the matrix is (pos or neg) definite.
if first_coeff > 0:
return "pos_def"
else:
return "neg_def"
def Watson_mass_at_2(self):
"""
Returns the local mass of the quadratic form when `p=2`, according
to Watson's Theorem 1 of "The 2-adic density of a quadratic form"
in Mathematika 23 (1976), pp 94--106.
INPUT:
none
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.Watson_mass_at_2() ## WARNING: WE NEED TO CHECK THIS CAREFULLY!
384
"""
## Make a 0-dim'l quadratic form (for initialization purposes)
Null_Form = copy.deepcopy(self)
Null_Form.__init__(ZZ, 0)
## Step 0: Compute Jordan blocks and bounds of the scales to keep track of
Jordan_Blocks = self.jordan_blocks_by_scale_and_unimodular(2)
scale_list = [B[0] for B in Jordan_Blocks]
s_min = min(scale_list)
s_max = max(scale_list)
## Step 1: Compute dictionaries of the diagonal block and 2x2 block for each scale
diag_dict = dict(
(i, Null_Form)
for i in range(s_min - 2, s_max + 4)) ## Initialize with the zero form
dim2_dict = dict(
(i, Null_Form)
for i in range(s_min, s_max + 4)) ## Initialize with the zero form
for (s, L) in Jordan_Blocks:
i = 0
while (i < L.dim() - 1) and (L[i, i + 1]
== 0): ## Find where the 2x2 blocks start
i = i + 1
if i < (L.dim() - 1):
diag_dict[s] = L.extract_variables(range(i)) ## Diagonal Form
dim2_dict[s + 1] = L.extract_variables(range(
i, L.dim())) ## Non-diagonal Form
else:
diag_dict[s] = L
#print "diag_dict = ", diag_dict
#print "dim2_dict = ", dim2_dict
#print "Jordan_Blocks = ", Jordan_Blocks
## Step 2: Compute three dictionaries of invariants (for n_j, m_j, nu_j)
n_dict = dict((j, 0) for j in range(s_min + 1, s_max + 2))
m_dict = dict((j, 0) for j in range(s_min, s_max + 4))
for (s, L) in Jordan_Blocks:
n_dict[s + 1] = L.dim()
if diag_dict[s].dim() == 0:
m_dict[s + 1] = ZZ.one() / ZZ(2) * L.dim()
else:
m_dict[s + 1] = floor(ZZ(L.dim() - 1) / ZZ(2))
#print " ==>", ZZ(L.dim() - 1) / ZZ(2), floor(ZZ(L.dim() - 1) / ZZ(2))
nu_dict = dict((j, n_dict[j + 1] - 2 * m_dict[j + 1])
for j in range(s_min, s_max + 1))
nu_dict[s_max + 1] = 0
#print "n_dict = ", n_dict
#print "m_dict = ", m_dict
#print "nu_dict = ", nu_dict
## Step 3: Compute the e_j dictionary
eps_dict = {}
for j in range(s_min, s_max + 3):
two_form = (diag_dict[j - 2] + diag_dict[j] +
dim2_dict[j]).scale_by_factor(2)
j_form = (two_form + diag_dict[j - 1]).base_change_to(
IntegerModRing(4))
if j_form.dim() == 0:
eps_dict[j] = 1
else:
iter_vec = [4] * j_form.dim()
alpha = sum([True for x in mrange(iter_vec) if j_form(x) == 0])
beta = sum([True for x in mrange(iter_vec) if j_form(x) == 2])
if alpha > beta:
eps_dict[j] = 1
elif alpha == beta:
eps_dict[j] = 0
else:
eps_dict[j] = -1
#print "eps_dict = ", eps_dict
## Step 4: Compute the quantities nu, q, P, E for the local mass at 2
nu = sum([j * n_dict[j] * (ZZ(n_dict[j] + 1) / ZZ(2) + \
sum([n_dict[r] for r in range(j+1, s_max+2)])) for j in range(s_min+1, s_max+2)])
q = sum([
sgn(nu_dict[j - 1] * (n_dict[j] + sgn(nu_dict[j])))
for j in range(s_min + 1, s_max + 2)
])
P = prod([
prod([1 - QQ(4)**(-i) for i in range(1, m_dict[j] + 1)])
for j in range(s_min + 1, s_max + 2)
])
E = prod([
ZZ(1) / ZZ(2) * (1 + eps_dict[j] * QQ(2)**(-m_dict[j]))
for j in range(s_min, s_max + 3)
])
#print "\nFinal Summary:"
#print "nu =", nu
#print "q = ", q
#print "P = ", P
#print "E = ", E
## Step 5: Compute the local mass for the prime 2.
mass_at_2 = QQ(2)**(nu - q) * P / E
return mass_at_2
def Watson_mass_at_2(self):
"""
Returns the local mass of the quadratic form when `p=2`, according
to Watson's Theorem 1 of "The 2-adic density of a quadratic form"
in Mathematika 23 (1976), pp 94--106.
INPUT:
none
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.Watson_mass_at_2() ## WARNING: WE NEED TO CHECK THIS CAREFULLY!
384
"""
## Make a 0-dim'l quadratic form (for initialization purposes)
Null_Form = copy.deepcopy(self)
Null_Form.__init__(ZZ, 0)
## Step 0: Compute Jordan blocks and bounds of the scales to keep track of
Jordan_Blocks = self.jordan_blocks_by_scale_and_unimodular(2)
scale_list = [B[0] for B in Jordan_Blocks]
s_min = min(scale_list)
s_max = max(scale_list)
## Step 1: Compute dictionaries of the diagonal block and 2x2 block for each scale
diag_dict = dict((i, Null_Form) for i in range(s_min-2, s_max + 4)) ## Initialize with the zero form
dim2_dict = dict((i, Null_Form) for i in range(s_min, s_max + 4)) ## Initialize with the zero form
for (s,L) in Jordan_Blocks:
i = 0
while (i < L.dim()-1) and (L[i,i+1] == 0): ## Find where the 2x2 blocks start
i = i + 1
if i < (L.dim() - 1):
diag_dict[s] = L.extract_variables(range(i)) ## Diagonal Form
dim2_dict[s+1] = L.extract_variables(range(i, L.dim())) ## Non-diagonal Form
else:
diag_dict[s] = L
#print "diag_dict = ", diag_dict
#print "dim2_dict = ", dim2_dict
#print "Jordan_Blocks = ", Jordan_Blocks
## Step 2: Compute three dictionaries of invariants (for n_j, m_j, nu_j)
n_dict = dict((j,0) for j in range(s_min+1, s_max+2))
m_dict = dict((j,0) for j in range(s_min, s_max+4))
for (s,L) in Jordan_Blocks:
n_dict[s+1] = L.dim()
if diag_dict[s].dim() == 0:
m_dict[s+1] = ZZ(1)/ZZ(2) * L.dim()
else:
m_dict[s+1] = floor(ZZ(L.dim() - 1) / ZZ(2))
#print " ==>", ZZ(L.dim() - 1) / ZZ(2), floor(ZZ(L.dim() - 1) / ZZ(2))
nu_dict = dict((j,n_dict[j+1] - 2*m_dict[j+1]) for j in range(s_min, s_max+1))
nu_dict[s_max+1] = 0
#print "n_dict = ", n_dict
#print "m_dict = ", m_dict
#print "nu_dict = ", nu_dict
## Step 3: Compute the e_j dictionary
eps_dict = {}
for j in range(s_min, s_max+3):
two_form = (diag_dict[j-2] + diag_dict[j] + dim2_dict[j]).scale_by_factor(2)
j_form = (two_form + diag_dict[j-1]).base_change_to(IntegerModRing(4))
if j_form.dim() == 0:
eps_dict[j] = 1
else:
iter_vec = [4] * j_form.dim()
alpha = sum([True for x in mrange(iter_vec) if j_form(x) == 0])
beta = sum([True for x in mrange(iter_vec) if j_form(x) == 2])
if alpha > beta:
eps_dict[j] = 1
elif alpha == beta:
eps_dict[j] = 0
else:
eps_dict[j] = -1
#print "eps_dict = ", eps_dict
## Step 4: Compute the quantities nu, q, P, E for the local mass at 2
nu = sum([j * n_dict[j] * (ZZ(n_dict[j] + 1) / ZZ(2) + \
sum([n_dict[r] for r in range(j+1, s_max+2)])) for j in range(s_min+1, s_max+2)])
q = sum([sgn(nu_dict[j-1] * (n_dict[j] + sgn(nu_dict[j]))) for j in range(s_min+1, s_max+2)])
P = prod([ prod([1 - QQ(4)**(-i) for i in range(1, m_dict[j]+1)]) for j in range(s_min+1, s_max+2)])
E = prod([ZZ(1)/ZZ(2) * (1 + eps_dict[j] * QQ(2)**(-m_dict[j])) for j in range(s_min, s_max+3)])
#print "\nFinal Summary:"
#print "nu =", nu
#print "q = ", q
#print "P = ", P
#print "E = ", E
## Step 5: Compute the local mass for the prime 2.
mass_at_2 = QQ(2)**(nu - q) * P / E
return mass_at_2
