def c3_func(SUK, prec=106):
r"""
Return the constant `c_3` from Smart's 1995 TCDF paper, [Sma1995]_
INPUT:
- ``SUK`` -- a group of `S`-units
- ``prec`` -- (default: 106) the precision of the real field
OUTPUT:
The constant ``c3``, as a real number
EXAMPLES::
sage: from sage.rings.number_field.S_unit_solver import c3_func
sage: K.<xi> = NumberField(x^3-3)
sage: SUK = UnitGroup(K, S=tuple(K.primes_above(3)))
sage: c3_func(SUK) # abs tol 1e-29
0.4257859134798034746197327286726
.. NOTE::
The numerator should be as close to 1 as possible, especially as the rank of the `S`-units grows large
REFERENCES:
- [Sma1995]_ p. 823
"""
R = RealField(prec)
all_places = list(SUK.primes()) + SUK.number_field().places(prec)
Possible_U = Combinations(all_places, SUK.rank())
c1 = R(0)
for U in Possible_U:
# first, build the matrix C_{i,U}
columns_of_C = []
for unit in SUK.fundamental_units():
columns_of_C.append(column_Log(SUK, unit, U, prec))
C = Matrix(SUK.rank(), SUK.rank(), columns_of_C)
# Is it invertible?
if abs(C.determinant()) > 10**(-10):
poss_c1 = C.inverse().apply_map(abs).norm(Infinity)
c1 = R(max(poss_c1, c1))
return R(0.9999999) / (c1*SUK.rank())
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
ALGORITHM:
Uses the PARI/GP function ``qfparam``.
EXAMPLES ::
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*x : -1/2*y + 1/2*z))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError("The conic self (=%s) is not smooth, hence does not have a parametrization." % self)
if point is None:
point = self.rational_point()
point = Sequence(point)
Q = PolynomialRing(QQ, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
M = self.symmetric_matrix()
M *= lcm([ t.denominator() for t in M.list() ])
par1 = qfparam(M, point)
B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
# self is in the image of B and does not lie on a line,
# hence B is invertible
A = B.inverse()
par2 = [sum([A[i,j]*gens[j] for j in range(3)]) for i in [1,0]]
par = ([Q(pol(x/y)*y**2) for pol in par1], par2)
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)
def automorphisms(self):
"""
Return a list of the automorphisms of the quadratic form.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.number_of_automorphisms() # optional -- souvigner
48
sage: 2^3 * factorial(3)
48
sage: len(Q.automorphisms())
48
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.number_of_automorphisms() # optional -- souvigner
16
sage: aut = Q.automorphisms()
sage: len(aut)
16
sage: print([Q(M) == Q for M in aut])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphisms()
[
[1 0 0] [-1 0 0]
[0 1 0] [ 0 -1 0]
[0 0 1], [ 0 0 -1]
]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.automorphisms()
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.automorphisms()
"""
## only for definite forms
if not self.is_definite():
raise ValueError("not a definite form in QuadraticForm.automorphisms()")
## Check for a cached value
try:
return self.__automorphisms
except AttributeError:
pass
## Find a basis of short vectors, and their lengths
basis, pivot_lengths = self.basis_of_short_vectors(show_lengths=True)
## List the relevant vectors by length
max_len = max(pivot_lengths)
vector_list_by_length = self.short_primitive_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis).transpose()
Ainv = A.inverse()
# A1 = A.inverse() * A.det()
# Q1 = A1.transpose() * self.matrix() * A1 ## This is the matrix of Q
# Q = self.matrix() * A.det()**2
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of Q in the new basis
Q3 = self.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
# ct = 0
## DIAGNOSTIC
# print "n = " + str(n)
# print "pivot_lengths = " + str(pivot_lengths)
# print "vector_list_by_length = " + str(vector_list_by_length)
# print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(vector_list_by_length[pivot_lengths[i]]) for i in range(n)]):
M = Matrix([vector_list_by_length[pivot_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
# Q1 = self.matrix()
# if self(M) == self:
# ct += 1
# print "ct = ", ct, " M = "
# print M
# print
if M.transpose() * Q3 * M == Q2: ## THIS DOES THE SAME THING! =(
Auto_list.append(M * Ainv)
## Cache the answer and return the list
self.__automorphisms = Auto_list
self.__number_of_automorphisms = len(Auto_list)
return Auto_list
def automorphisms(self):
"""
Return a list of the automorphisms of the quadratic form.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.number_of_automorphisms() # optional -- souvigner
48
sage: 2^3 * factorial(3)
48
sage: len(Q.automorphisms())
48
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.number_of_automorphisms() # optional -- souvigner
16
sage: aut = Q.automorphisms()
sage: len(aut)
16
sage: print([Q(M) == Q for M in aut])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphisms()
[
[1 0 0] [-1 0 0]
[0 1 0] [ 0 -1 0]
[0 0 1], [ 0 0 -1]
]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.automorphisms()
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.automorphisms()
"""
## only for definite forms
if not self.is_definite():
raise ValueError("not a definite form in QuadraticForm.automorphisms()")
## Check for a cached value
try:
return self.__automorphisms
except AttributeError:
pass
## Find a basis of short vectors, and their lengths
basis, pivot_lengths = self.basis_of_short_vectors(show_lengths=True)
## List the relevant vectors by length
max_len = max(pivot_lengths)
vector_list_by_length = self.short_primitive_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis).transpose()
Ainv = A.inverse()
#A1 = A.inverse() * A.det()
#Q1 = A1.transpose() * self.matrix() * A1 ## This is the matrix of Q
#Q = self.matrix() * A.det()**2
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of Q in the new basis
Q3 = self.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
#ct = 0
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(vector_list_by_length[pivot_lengths[i]]) for i in range(n)]):
M = Matrix([vector_list_by_length[pivot_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
#Q1 = self.matrix()
#if self(M) == self:
#ct += 1
#print "ct = ", ct, " M = "
#print M
#print
if M.transpose() * Q3 * M == Q2: ## THIS DOES THE SAME THING! =(
Auto_list.append(M * Ainv)
## Cache the answer and return the list
self.__automorphisms = Auto_list
self.__number_of_automorphisms = len(Auto_list)
return Auto_list
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
ALGORITHM:
Uses the PARI/GP function ``qfparam``.
EXAMPLES ::
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*x : -1/2*y + 1/2*z))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError(
"The conic self (=%s) is not smooth, hence does not have a parametrization."
% self)
if point is None:
point = self.rational_point()
point = Sequence(point)
Q = PolynomialRing(QQ, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
M = self.symmetric_matrix()
M *= lcm([t.denominator() for t in M.list()])
par1 = qfparam(M, point)
B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
# self is in the image of B and does not lie on a line,
# hence B is invertible
A = B.inverse()
par2 = [sum([A[i, j] * gens[j] for j in range(3)]) for i in [1, 0]]
par = ([Q(pol(x / y) * y**2) for pol in par1], par2)
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0], self), self.Hom(P1)(par[1], check=False)
def is_globally_equivalent_to(self,
other,
return_matrix=False,
check_theta_to_precision='sturm',
check_local_equivalence=True):
"""
Determines if the current quadratic form is equivalent to the
given form over ZZ. If return_matrix is True, then we also return
the transformation matrix M so that self(M) == other.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.(Q1) # optional -- souvigner
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True) # optional -- souvigner
sage: Q(MM) == Q1 # optional -- souvigner
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_globally_equivalent_to(Q3) # optional -- souvigner
True
sage: M = Q1.is_globally_equivalent_to(Q3, True) ; M # optional -- souvigner
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3 # optional -- souvigner
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
"""
## only for definite forms
if not self.is_definite():
raise ValueError(
"not a definite form in QuadraticForm.is_globally_equivalent_to()")
## Check that other is a QuadraticForm
#if not isinstance(other, QuadraticForm):
if not is_QuadraticForm(other):
raise TypeError(
"Oops! You must compare two quadratic forms, but the argument is not a quadratic form. =("
)
## Now use the Souvigner code by default! =)
return other.is_globally_equivalent__souvigner(
self, return_matrix
) ## Note: We switch this because the Souvigner code has the opposite mapping convention to us. (It takes the second argument to the first!)
## ---------------------------------- Unused Code below ---------------------------------------------------------
## Check if the forms are locally equivalent
if (check_local_equivalence == True):
if not self.is_locally_equivalent_to(other):
return False
## Check that the forms have the same theta function up to the desired precision (this can be set so that it determines the cusp form)
if check_theta_to_precision is not None:
if self.theta_series(
check_theta_to_precision, var_str='',
safe_flag=False) != other.theta_series(
check_theta_to_precision, var_str='', safe_flag=False):
return False
## Make all possible matrices which give an isomorphism -- can we do this more intelligently?
## ------------------------------------------------------------------------------------------
## Find a basis of short vectors for one form, and try to match them with vectors of that length in the other one.
basis_for_self, self_lengths = self.basis_of_short_vectors(
show_lengths=True)
max_len = max(self_lengths)
short_vectors_of_other = other.short_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis_for_self).transpose()
Q2 = A.transpose() * self.matrix(
) * A ## This is the matrix of 'self' in the new basis
Q3 = other.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange(
[len(short_vectors_of_other[self_lengths[i]]) for i in range(n)]):
M = Matrix([
short_vectors_of_other[self_lengths[i]][index_vec[i]]
for i in range(n)
]).transpose()
if M.transpose() * Q3 * M == Q2:
if return_matrix:
return A * M.inverse()
else:
return True
## If we got here, then there is no isomorphism
return False
def is_globally_equivalent_to(self, other, return_matrix=False, check_theta_to_precision='sturm', check_local_equivalence=True):
"""
Determines if the current quadratic form is equivalent to the
given form over ZZ. If return_matrix is True, then we also return
the transformation matrix M so that self(M) == other.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.(Q1) # optional -- souvigner
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True) # optional -- souvigner
sage: Q(MM) == Q1 # optional -- souvigner
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_globally_equivalent_to(Q3) # optional -- souvigner
True
sage: M = Q1.is_globally_equivalent_to(Q3, True) ; M # optional -- souvigner
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3 # optional -- souvigner
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
"""
## only for definite forms
if not self.is_definite():
raise ValueError, "not a definite form in QuadraticForm.is_globally_equivalent_to()"
## Check that other is a QuadraticForm
#if not isinstance(other, QuadraticForm):
if not is_QuadraticForm(other):
raise TypeError, "Oops! You must compare two quadratic forms, but the argument is not a quadratic form. =("
## Now use the Souvigner code by default! =)
return other.is_globally_equivalent__souvigner(self, return_matrix) ## Note: We switch this because the Souvigner code has the opposite mapping convention to us. (It takes the second argument to the first!)
## ---------------------------------- Unused Code below ---------------------------------------------------------
## Check if the forms are locally equivalent
if (check_local_equivalence == True):
if not self.is_locally_equivalent_to(other):
return False
## Check that the forms have the same theta function up to the desired precision (this can be set so that it determines the cusp form)
if check_theta_to_precision != None:
if self.theta_series(check_theta_to_precision, var_str='', safe_flag=False) != other.theta_series(check_theta_to_precision, var_str='', safe_flag=False):
return False
## Make all possible matrices which give an isomorphism -- can we do this more intelligently?
## ------------------------------------------------------------------------------------------
## Find a basis of short vectors for one form, and try to match them with vectors of that length in the other one.
basis_for_self, self_lengths = self.basis_of_short_vectors(show_lengths=True)
max_len = max(self_lengths)
short_vectors_of_other = other.short_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis_for_self).transpose()
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of 'self' in the new basis
Q3 = other.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(short_vectors_of_other[self_lengths[i]]) for i in range(n)]):
M = Matrix([short_vectors_of_other[self_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
if M.transpose() * Q3 * M == Q2:
if return_matrix:
return A * M.inverse()
else:
return True
## If we got here, then there is no isomorphism
return False
