def complementary_subform_to_vector(self, v):
"""
Finds the `(n-1)`-dim'l quadratic form orthogonal to the vector `v`.
Note: This is usually not a direct summand!
Technical Notes: There is a minor difference in the cancellation
code here (form the C++ version) since the notation Q `[i,j]` indexes
coefficients of the quadratic polynomial here, not the symmetric
matrix. Also, it produces a better splitting now, for the full
lattice (as opposed to a sublattice in the C++ code) since we
now extend `v` to a unimodular matrix.
INPUT:
`v` -- a list of self.dim() integers
OUTPUT:
a QuadraticForm over `ZZ`
EXAMPLES::
sage: Q1 = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q1.complementary_subform_to_vector([1,0,0,0])
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 0 0 ]
[ * 5 0 ]
[ * * 7 ]
::
sage: Q1.complementary_subform_to_vector([1,1,0,0])
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 12 0 0 ]
[ * 5 0 ]
[ * * 7 ]
::
sage: Q1.complementary_subform_to_vector([1,1,1,1])
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 624 -480 -672 ]
[ * 880 -1120 ]
[ * * 1008 ]
"""
n = self.dim()
## Copy the quadratic form
Q = deepcopy(self)
## Find the first non-zero component of v, and call it nz (Note: 0 <= nz < n)
nz = 0
while (nz < n) and (v[nz] == 0):
nz += 1
## Abort if v is the zero vector
if nz == n:
raise TypeError("Oops, v cannot be the zero vector! =(")
## Make the change of basis matrix
new_basis = extend_to_primitive(matrix(ZZ, n, 1, v))
## Change Q (to Q1) to have v as its nz-th basis vector
Q1 = Q(new_basis)
## Pick out the value Q(v) of the vector
d = Q1[0, 0]
## For each row/column, perform elementary operations to cancel them out.
for i in range(1, n):
## Check if the (i,0)-entry is divisible by d,
## and stretch its row/column if not.
if Q1[i, 0] % d != 0:
Q1 = Q1.multiply_variable(d / GCD(d, Q1[i, 0] // 2), i)
## Now perform the (symmetric) elementary operations to cancel out the (i,0) entries/
Q1 = Q1.add_symmetric(-(Q1[i, 0] / 2) / (GCD(d, Q1[i, 0] // 2)), i, 0)
## Check that we're done!
done_flag = True
for i in range(1, n):
if Q1[0, i] != 0:
done_flag = False
if not done_flag:
raise RuntimeError(
"There is a problem cancelling out the matrix entries! =O")
## Return the complementary matrix
return Q1.extract_variables(range(1, n))
def complementary_subform_to_vector(self, v):
"""
Finds the `(n-1)`-dim'l quadratic form orthogonal to the vector `v`.
Note: This is usually not a direct summand!
Technical Notes: There is a minor difference in the cancellation
code here (form the C++ version) since the notation Q `[i,j]` indexes
coefficients of the quadratic polynomial here, not the symmetric
matrix. Also, it produces a better splitting now, for the full
lattice (as opposed to a sublattice in the C++ code) since we
now extend `v` to a unimodular matrix.
INPUT:
`v` -- a list of self.dim() integers
OUTPUT:
a QuadraticForm over `ZZ`
EXAMPLES::
sage: Q1 = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q1.complementary_subform_to_vector([1,0,0,0])
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 0 0 ]
[ * 5 0 ]
[ * * 7 ]
::
sage: Q1.complementary_subform_to_vector([1,1,0,0])
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 12 0 0 ]
[ * 5 0 ]
[ * * 7 ]
::
sage: Q1.complementary_subform_to_vector([1,1,1,1])
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 624 -480 -672 ]
[ * 880 -1120 ]
[ * * 1008 ]
"""
n = self.dim()
## Copy the quadratic form
Q = deepcopy(self)
## Find the first non-zero component of v, and call it nz (Note: 0 <= nz < n)
nz = 0
while (nz < n) and (v[nz] == 0):
nz += 1
## Abort if v is the zero vector
if nz == n:
raise TypeError("Oops, v cannot be the zero vector! =(")
## Make the change of basis matrix
new_basis = extend_to_primitive(matrix(ZZ,n,1,v))
## Change Q (to Q1) to have v as its nz-th basis vector
Q1 = Q(new_basis)
## Pick out the value Q(v) of the vector
d = Q1[0, 0]
#print Q1
## For each row/column, perform elementary operations to cancel them out.
for i in range(1,n):
## Check if the (i,0)-entry is divisible by d,
## and stretch its row/column if not.
if Q1[i,0] % d != 0:
Q1 = Q1.multiply_variable(d / GCD(d, Q1[i, 0]//2), i)
#print "After scaling at i =", i
#print Q1
## Now perform the (symmetric) elementary operations to cancel out the (i,0) entries/
Q1 = Q1.add_symmetric(-(Q1[i,0]/2) / (GCD(d, Q1[i,0]//2)), i, 0)
#print "After cancelling at i =", i
#print Q1
## Check that we're done!
done_flag = True
for i in range(1, n):
if Q1[0,i] != 0:
done_flag = False
if done_flag == False:
raise RuntimeError("There is a problem cancelling out the matrix entries! =O")
## Return the complementary matrix
return Q1.extract_variables(range(1,n))
def find_p_neighbor_from_vec(self, p, v):
"""
Finds the `p`-neighbor of this quadratic form associated to a given
vector `v` satisfying:
#. `Q(v) = 0 \pmod p`
#. `v` is a non-singular point of the conic `Q(v) = 0 \pmod p`.
Reference: Gonzalo Tornaria's Thesis, Thrm 3.5, p34.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ,[1,1,1,1])
sage: v = vector([0,2,1,1])
sage: X = Q.find_p_neighbor_from_vec(3,v); X
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 3 10 0 -4 ]
[ * 9 0 -6 ]
[ * * 1 0 ]
[ * * * 2 ]
"""
R = self.base_ring()
n = self.dim()
B2 = self.matrix()
## Find a (dual) vector w with B(v,w) != 0 (mod p)
v_dual = B2 * vector(v) ## We want the dot product with this to not be divisible by 2*p.
y_ind = 0
while ((y_ind < n) and (v_dual[y_ind] % p) == 0): ## Check the dot product for the std basis vectors!
y_ind += 1
if y_ind == n:
raise RuntimeError("Oops! One of the standard basis vectors should have worked.")
w = vector([R(i == y_ind) for i in range(n)])
vw_prod = (v * self.matrix()).dot_product(w)
## DIAGNOSTIC
#if vw_prod == 0:
# print "v = ", v
# print "v_dual = ", v_dual
# print "v_dual[y_ind] = ", v_dual[y_ind]
# print "(v_dual[y_ind] % p) = ", (v_dual[y_ind] % p)
# print "w = ", w
# print "p = ", p
# print "vw_prod = ", vw_prod
# raise RuntimeError, "ERROR: Why is vw_prod = 0?"
## DIAGNOSTIC
#print "v = ", v
#print "w = ", w
#print "vw_prod = ", vw_prod
## Lift the vector v to a vector v1 s.t. Q(v1) = 0 (mod p^2)
s = self(v)
if (s % p**2 != 0):
al = (-s / (p * vw_prod)) % p
v1 = v + p * al * w
v1w_prod = (v1 * self.matrix()).dot_product(w)
else:
v1 = v
v1w_prod = vw_prod
## DIAGNOSTIC
#if (s % p**2 != 0):
# print "al = ", al
#print "v1 = ", v1
#print "v1w_prod = ", v1w_prod
## Construct a special p-divisible basis to use for the p-neighbor switch
good_basis = extend_to_primitive([v1, w])
for i in range(2,n):
ith_prod = (good_basis[i] * self.matrix()).dot_product(v)
c = (ith_prod / v1w_prod) % p
good_basis[i] = good_basis[i] - c * w ## Ensures that this extension has <v_i, v> = 0 (mod p)
## DIAGNOSTIC
#print "original good_basis = ", good_basis
## Perform the p-neighbor switch
good_basis[0] = vector([x/p for x in good_basis[0]]) ## Divide v1 by p
good_basis[1] = good_basis[1] * p ## Multiply w by p
## Return the associated quadratic form
M = matrix(good_basis)
new_Q = deepcopy(self) ## Note: This avoids a circular import of QuadraticForm!
new_Q.__init__(R, M * self.matrix() * M.transpose())
return new_Q
return QuadraticForm(R, M * self.matrix() * M.transpose())
def find_p_neighbor_from_vec(self, p, v):
"""
Finds the `p`-neighbor of this quadratic form associated to a given
vector `v` satisfying:
#. `Q(v) = 0 \pmod p`
#. `v` is a non-singular point of the conic `Q(v) = 0 \pmod p`.
Reference: Gonzalo Tornaria's Thesis, Thrm 3.5, p34.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ,[1,1,1,1])
sage: v = vector([0,2,1,1])
sage: X = Q.find_p_neighbor_from_vec(3,v); X
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 3 10 0 -4 ]
[ * 9 0 -6 ]
[ * * 1 0 ]
[ * * * 2 ]
"""
R = self.base_ring()
n = self.dim()
B2 = self.matrix()
## Find a (dual) vector w with B(v,w) != 0 (mod p)
v_dual = B2 * vector(
v) ## We want the dot product with this to not be divisible by 2*p.
y_ind = 0
while ((y_ind < n) and (v_dual[y_ind] % p)
== 0): ## Check the dot product for the std basis vectors!
y_ind += 1
if y_ind == n:
raise RuntimeError(
"Oops! One of the standard basis vectors should have worked.")
w = vector([R(i == y_ind) for i in range(n)])
vw_prod = (v * self.matrix()).dot_product(w)
## DIAGNOSTIC
#if vw_prod == 0:
# print "v = ", v
# print "v_dual = ", v_dual
# print "v_dual[y_ind] = ", v_dual[y_ind]
# print "(v_dual[y_ind] % p) = ", (v_dual[y_ind] % p)
# print "w = ", w
# print "p = ", p
# print "vw_prod = ", vw_prod
# raise RuntimeError, "ERROR: Why is vw_prod = 0?"
## DIAGNOSTIC
#print "v = ", v
#print "w = ", w
#print "vw_prod = ", vw_prod
## Lift the vector v to a vector v1 s.t. Q(v1) = 0 (mod p^2)
s = self(v)
if (s % p**2 != 0):
al = (-s / (p * vw_prod)) % p
v1 = v + p * al * w
v1w_prod = (v1 * self.matrix()).dot_product(w)
else:
v1 = v
v1w_prod = vw_prod
## DIAGNOSTIC
#if (s % p**2 != 0):
# print "al = ", al
#print "v1 = ", v1
#print "v1w_prod = ", v1w_prod
## Construct a special p-divisible basis to use for the p-neighbor switch
good_basis = extend_to_primitive([v1, w])
for i in range(2, n):
ith_prod = (good_basis[i] * self.matrix()).dot_product(v)
c = (ith_prod / v1w_prod) % p
good_basis[i] = good_basis[
i] - c * w ## Ensures that this extension has <v_i, v> = 0 (mod p)
## DIAGNOSTIC
#print "original good_basis = ", good_basis
## Perform the p-neighbor switch
good_basis[0] = vector([x / p for x in good_basis[0]]) ## Divide v1 by p
good_basis[1] = good_basis[1] * p ## Multiply w by p
## Return the associated quadratic form
M = matrix(good_basis)
new_Q = deepcopy(
self) ## Note: This avoids a circular import of QuadraticForm!
new_Q.__init__(R, M * self.matrix() * M.transpose())
return new_Q
return QuadraticForm(R, M * self.matrix() * M.transpose())
def find_p_neighbor_from_vec(self, p, v):
r"""
Find the `p`-neighbor of this quadratic form associated to a given
vector `v` satisfying:
#. `Q(v) = 0 \pmod p`
#. `v` is a non-singular point of the conic `Q(v) = 0 \pmod p`.
Reference: Gonzalo Tornaria's Thesis, Theorem 3.5, p34.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ,[1,1,1,1])
sage: v = vector([0,2,1,1])
sage: X = Q.find_p_neighbor_from_vec(3,v); X
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 3 10 0 -4 ]
[ * 9 0 -6 ]
[ * * 1 0 ]
[ * * * 2 ]
"""
R = self.base_ring()
n = self.dim()
B2 = self.matrix()
# Find a (dual) vector w with B(v,w) != 0 (mod p)
# We want the dot product with this to not be divisible by 2*p.
v_dual = B2 * vector(v)
y_ind = 0
while ((y_ind < n) and (v_dual[y_ind] % p) == 0):
# Check the dot product for the std basis vectors!
y_ind += 1
if y_ind == n:
raise RuntimeError("Oops! One of the standard basis vectors "
"should have worked.")
w = vector(R, [R(i == y_ind) for i in range(n)])
vw_prod = (v * self.matrix()).dot_product(w)
# Lift the vector v to a vector v1 s.t. Q(v1) = 0 (mod p^2)
s = self(v)
if (s % p**2 != 0):
al = (-s / (p * vw_prod)) % p
v1 = v + p * al * w
v1w_prod = (v1 * self.matrix()).dot_product(w)
else:
v1 = v
v1w_prod = vw_prod
# Construct a special p-divisible basis to use for the p-neighbor switch
good_basis = extend_to_primitive([v1, w])
for i in range(2, n):
ith_prod = (good_basis[i] * self.matrix()).dot_product(v)
c = (ith_prod / v1w_prod) % p
good_basis[i] = good_basis[i] - c * w
# Ensures that this extension has <v_i, v> = 0 (mod p)
# Perform the p-neighbor switch
good_basis[0] = vector([x / p for x in good_basis[0]]) # Divide v1 by p
good_basis[1] = good_basis[1] * p # Multiply w by p
# Return the associated quadratic form
M = matrix(good_basis)
QF = self.parent()
return QF(R, M * self.matrix() * M.transpose())