def __init__(self, n):
"""
Initialize ``self``.
EXAMPLES::
sage: G = groups.matrix.BinaryDihedral(4)
sage: TestSuite(G).run()
"""
self._n = n
if n % 2 == 0:
R = CyclotomicField(2 * n)
zeta = R.gen()
i = R.gen()**(n // 2)
else:
R = CyclotomicField(4 * n)
zeta = R.gen()**2
i = R.gen()**n
MS = MatrixSpace(R, 2)
zero = R.zero()
gens = [MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero])]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self,
ZZ(2),
R,
gap_group,
category=Groups().Finite())
def __init__(self, n):
"""
Initialize ``self``.
EXAMPLES::
sage: G = groups.matrix.BinaryDihedral(4)
sage: TestSuite(G).run()
"""
self._n = n
if n % 2 == 0:
R = CyclotomicField(2*n)
zeta = R.gen()
i = R.gen()**(n//2)
else:
R = CyclotomicField(4*n)
zeta = R.gen()**2
i = R.gen()**n
MS = MatrixSpace(R, 2)
zero = R.zero()
gens = [ MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero]) ]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self, ZZ(2), R, gap_group, category=Groups().Finite())
def to_matrix(self):
"""
Return a matrix of ``self``.
The colors are mapped to roots of unity.
EXAMPLES::
sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t*s2; x.one_line_form()
[(1, 2), (0, 1), (0, 3)]
sage: M = x.to_matrix(); M
[ 0 1 0]
[zeta4 0 0]
[ 0 0 1]
The matrix multiplication is in the *opposite* order::
sage: M == s2.to_matrix()*t.to_matrix()*s2.to_matrix()*s1.to_matrix()
True
"""
Cp = CyclotomicField(self.parent()._m)
g = Cp.gen()
D = diagonal_matrix(Cp, [g**i for i in self._colors])
return self._perm.to_matrix() * D
def to_matrix(self):
"""
Return a matrix of ``self``.
The colors are mapped to roots of unity.
EXAMPLES::
sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t*s2; x.one_line_form()
[(1, 2), (0, 1), (0, 3)]
sage: M = x.to_matrix(); M
[ 0 1 0]
[zeta4 0 0]
[ 0 0 1]
The matrix multiplication is in the *opposite* order::
sage: M == s2.to_matrix()*t.to_matrix()*s2.to_matrix()*s1.to_matrix()
True
"""
Cp = CyclotomicField(self.parent()._m)
g = Cp.gen()
D = diagonal_matrix(Cp, [g ** i for i in self._colors])
return self._perm.to_matrix() * D
def _calcMatrixTrans(calc, tS, tT, lS, lT): """ See `calcMatrixTrans()` for the main documentation. This is the lower-level function which is wrapped through `persistent_cache`. This makes the cache more flexible about different parameters `R` to `calcMatrixTrans()`. """ try: ms = calc.calcMatrixTrans(tS, tT, lS, lT) except Exception: print (calc.params, calc.curlS, tS, tT, lS, lT) raise # Each matrix is for a zeta**i factor, where zeta is the n-th root of unity. # And n = calc.matrixCountTrans. assert len(ms) == calc.matrixCountTrans order = len(ms) K = CyclotomicField(order) zeta = K.gen() Kcoords = zeta.coordinates_in_terms_of_powers() assert len(K.power_basis()) == K.degree() new_ms = [matrix(QQ, ms[0].nrows(), ms[0].ncols()) for i in range(K.degree())] for l in range(order): coords = Kcoords(zeta**l) for i,m in enumerate(coords): new_ms[i] += ms[l] * m ms = new_ms denom = calc.matrixRowDenomTrans denom, ms = reduceNRow(denom=denom, mats=ms) return denom, order, ms
def to_cyclotomic_field(self, R=None):
r"""
Return this element as an element of a cyclotomic field.
EXAMPLES::
sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(3).to_cyclotomic_field()
zeta3
sage: UCF.gen(3,2).to_cyclotomic_field()
-zeta3 - 1
sage: CF = CyclotomicField(5)
sage: CF(E(5)) # indirect doctest
zeta5
sage: CF = CyclotomicField(7)
sage: CF(E(5)) # indirect doctest
Traceback (most recent call last):
...
TypeError: Cannot coerce zeta5 into Cyclotomic Field of order 7 and
degree 6
sage: CF = CyclotomicField(10)
sage: CF(E(5)) # indirect doctest
zeta10^2
Matrices are correctly dealt with::
sage: M = Matrix(UCF,2,[E(3),E(4),E(5),E(6)]); M
[ E(3) E(4)]
[ E(5) -E(3)^2]
sage: Matrix(CyclotomicField(60),M) # indirect doctest
[zeta60^10 - 1 zeta60^15]
[ zeta60^12 zeta60^10]
Using a non-standard embedding::
sage: CF = CyclotomicField(5,embedding=CC(exp(4*pi*i/5)))
sage: x = E(5)
sage: CC(x)
0.309016994374947 + 0.951056516295154*I
sage: CC(CF(x))
0.309016994374947 + 0.951056516295154*I
"""
from sage.rings.number_field.number_field import CyclotomicField
k = self._obj.Conductor().sage()
Rcan = CyclotomicField(k)
if R is None:
R = Rcan
obj = self._obj
if obj.IsRat():
return R(obj.sage())
zeta = Rcan.gen()
coeffs = obj.CoeffsCyc(k).sage()
return R(sum(coeffs[a] * zeta**a for a in range(1,k)))
def to_cyclotomic_field(self, R=None):
r"""
Return this element as an element of a cyclotomic field.
EXAMPLES::
sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(3).to_cyclotomic_field()
zeta3
sage: UCF.gen(3,2).to_cyclotomic_field()
-zeta3 - 1
sage: CF = CyclotomicField(5)
sage: CF(E(5)) # indirect doctest
zeta5
sage: CF = CyclotomicField(7)
sage: CF(E(5)) # indirect doctest
Traceback (most recent call last):
...
TypeError: Cannot coerce zeta5 into Cyclotomic Field of order 7 and
degree 6
sage: CF = CyclotomicField(10)
sage: CF(E(5)) # indirect doctest
zeta10^2
Matrices are correctly dealt with::
sage: M = Matrix(UCF,2,[E(3),E(4),E(5),E(6)]); M
[ E(3) E(4)]
[ E(5) -E(3)^2]
sage: Matrix(CyclotomicField(60),M) # indirect doctest
[zeta60^10 - 1 zeta60^15]
[ zeta60^12 zeta60^10]
Using a non-standard embedding::
sage: CF = CyclotomicField(5,embedding=CC(exp(4*pi*i/5)))
sage: x = E(5)
sage: CC(x)
0.309016994374947 + 0.951056516295154*I
sage: CC(CF(x))
0.309016994374947 + 0.951056516295154*I
"""
from sage.rings.number_field.number_field import CyclotomicField
k = self._obj.Conductor().sage()
Rcan = CyclotomicField(k)
if R is None:
R = Rcan
obj = self._obj
if obj.IsRat():
return R(obj.sage())
zeta = Rcan.gen()
coeffs = obj.CoeffsCyc(k).sage()
return R(sum(coeffs[a] * zeta**a for a in range(1, k)))
def _rank(self, K) :
if K is QQ or K in NumberFields() :
return len(_jacobi_forms_by_taylor_expansion_coords(self.__index, self.__weight, 0))
## This is the formula used by Poor and Yuen in Paramodular cusp forms
if self.__weight == 2 :
delta = len(self.__index.divisors()) // 2 - 1
else :
delta = 0
return sum( ModularForms(1, self.__weight + 2 * j).dimension() + j**2 // (4 * self.__index)
for j in xrange(self.__index + 1) ) \
+ delta
## This is the formula given by Skoruppa in
## Jacobi forms of critical weight and Weil representations
##FIXME: There is some mistake here
if self.__weight % 2 != 0 :
## Otherwise the space X(i**(n - 2 k)) is different
## See: Skoruppa, Jacobi forms of critical weight and Weil representations
raise NotImplementedError
m = self.__index
K = CyclotomicField(24 * m, 'zeta')
zeta = K.gen(0)
quadform = lambda x : 6 * x**2
bilinform = lambda x,y : quadform(x + y) - quadform(x) - quadform(y)
T = diagonal_matrix([zeta**quadform(i) for i in xrange(2*m)])
S = sum(zeta**(-quadform(x)) for x in xrange(2 * m)) / (2 * m) \
* matrix([[zeta**(-bilinform(j,i)) for j in xrange(2*m)] for i in xrange(2*m)])
subspace_matrix_1 = matrix( [ [1 if j == i or j == 2*m - i else 0 for j in xrange(m + 1) ]
for i in xrange(2*m)] )
subspace_matrix_2 = zero_matrix(ZZ, m + 1, 2*m)
subspace_matrix_2.set_block(0,0,identity_matrix(m+1))
T = subspace_matrix_2 * T * subspace_matrix_1
S = subspace_matrix_2 * S * subspace_matrix_1
sqrt3 = (zeta**(4*m) - zeta**(-4*m)) * zeta**(-6*m)
rank = (self.__weight - 1/2 - 1) / 2 * (m + 1) \
+ 1/8 * ( zeta**(3*m * (2*self.__weight - 1)) * S.trace()
+ zeta**(3*m * (1 - 2*self.__weight)) * S.trace().conjugate() ) \
+ 2/(3*sqrt3) * ( zeta**(4 * m * self.__weight) * (S*T).trace()
+ zeta**(-4 * m * self.__weight) * (S*T).trace().conjugate() ) \
- sum((j**2 % (m+1))/(m+1) -1/2 for j in range(0,m+1))
if self.__weight > 5 / 2 :
return rank
else :
raise NotImplementedError
raise NotImplementedError
def sage_character_to_magma(chi,N=None,magma=None):
if magma is None:
magma = sage.interfaces.magma
if N is None:
N = chi.modulus()
else:
N = ZZ(N)
chi = chi.extend(N)
G = chi.parent()
order = chi.order()
gens = G.unit_gens()
ElGm = magma.DirichletGroupFull(N)
Kcyc = CyclotomicField(ElGm.BaseRing().CyclotomicOrder(),names='z')
phi = ElGm.BaseRing().sage().hom([Kcyc.gen()])
target = [chi(g) for g in gens]
for chim in ElGm.Elements():
if chim.Order().sage() == order:
this = [phi(chim.Evaluate(g).sage()) for g in gens]
if all((u == v for u, v in zip(this, target))):
return chim
raise RuntimeError("Should not get to this point")
def toCyclPowerBase(M, order):
"""
Let's
K = CyclotomicField(order).
INPUT:
- `M` -- A matrix over the cyclomotic field `K`.
- `order` -- The order of `K`, the cyclomotic field.
OUTPUT:
- A list of matrices `ms` in power base where every matrix
is a factor to `zeta**i` where `zeta = K.gen()`
and `len(ms) == K.degree()`.
"""
K = CyclotomicField(order)
zeta = K.gen()
Kcoords = zeta.coordinates_in_terms_of_powers()
assert len(K.power_basis()) == K.degree()
ms = [matrix(QQ,M.nrows(),M.ncols()) for i in range(K.degree())]
for y in range(M.nrows()):
for x in range(M.ncols()):
try:
v_ = M[y,x]
v = K(v_)
coords = Kcoords(v)
except TypeError:
print "type of {1} ({2}) is not valid in Cyclomotic field of order {0}".format(order, M[y,x], type(M[y,x]))
raise
assert len(coords) == K.degree()
for i in range(K.degree()):
ms[i][y,x] = coords[i]
return ms
def molien_series(self, chi=None, return_series=True, prec=20, variable='t'):
r"""
Compute the Molien series of this finite group with respect to the
character ``chi``. It can be returned either as a rational function
in one variable or a power series in one variable. The base field
must be a finite field, the rationals, or a cyclotomic field.
Note that the base field characteristic cannot divide the group
order (i.e., the non-modular case).
ALGORITHM:
For a finite group `G` in characteristic zero we construct the Molien series as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\text{det}(I-tg)},
where `I` is the identity matrix and `t` an indeterminate.
For characteristic `p` not dividing the order of `G`, let `k` be the base field
and `N` the order of `G`. Define `\lambda` as a primitive `N`-th root of unity over `k`
and `\omega` as a primitive `N`-th root of unity over `\QQ`. For each `g \in G`
define `k_i(g)` to be the positive integer such that
`e_i = \lambda^{k_i(g)}` for each eigenvalue `e_i` of `g`. Then the Molien series
is computed as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\prod_{i=1}^n(1 - t\omega^{k_i(g)})},
where `t` is an indeterminant. [Dec1998]_
INPUT:
- ``chi`` -- (default: trivial character) a linear group character of this group
- ``return_series`` -- boolean (default: ``True``) if ``True``, then returns
the Molien series as a power series, ``False`` as a rational function
- ``prec`` -- integer (default: 20); power series default precision
- ``variable`` -- string (default: ``'t'``); Variable name for the Molien series
OUTPUT: single variable rational function or power series with integer coefficients
EXAMPLES::
sage: MatrixGroup(matrix(QQ,2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups
sage: MatrixGroup(matrix(GF(3),2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: characteristic cannot divide group order
Tetrahedral Group::
sage: K.<i> = CyclotomicField(4)
sage: Tetra = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [0,i, -i,0])
sage: Tetra.molien_series(prec=30)
1 + t^8 + 2*t^12 + t^16 + 2*t^20 + 3*t^24 + 2*t^28 + O(t^30)
sage: mol = Tetra.molien_series(return_series=False); mol
(t^8 - t^4 + 1)/(t^16 - t^12 - t^4 + 1)
sage: mol.parent()
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
sage: chi = Tetra.character(Tetra.character_table()[1])
sage: Tetra.molien_series(chi, prec=30, variable='u')
u^6 + u^14 + 2*u^18 + u^22 + 2*u^26 + 3*u^30 + 2*u^34 + O(u^36)
sage: chi = Tetra.character(Tetra.character_table()[2])
sage: Tetra.molien_series(chi)
t^10 + t^14 + t^18 + 2*t^22 + 2*t^26 + O(t^30)
::
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: mol = S3.molien_series(prec=10); mol
1 + t + 2*t^2 + 3*t^3 + 4*t^4 + 5*t^5 + 7*t^6 + 8*t^7 + 10*t^8 + 12*t^9 + O(t^10)
sage: mol.parent()
Power Series Ring in t over Integer Ring
Octahedral Group::
sage: K.<v> = CyclotomicField(8)
sage: a = v-v^3 #sqrt(2)
sage: i = v^2
sage: Octa = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [(1+i)/a,0, 0,(1-i)/a])
sage: Octa.molien_series(prec=30)
1 + t^8 + t^12 + t^16 + t^18 + t^20 + 2*t^24 + t^26 + t^28 + O(t^30)
Icosahedral Group::
sage: K.<v> = CyclotomicField(10)
sage: z5 = v^2
sage: i = z5^5
sage: a = 2*z5^3 + 2*z5^2 + 1 #sqrt(5)
sage: Ico = MatrixGroup([[z5^3,0, 0,z5^2], [0,1, -1,0], [(z5^4-z5)/a, (z5^2-z5^3)/a, (z5^2-z5^3)/a, -(z5^4-z5)/a]])
sage: Ico.molien_series(prec=40)
1 + t^12 + t^20 + t^24 + t^30 + t^32 + t^36 + O(t^40)
::
sage: G = MatrixGroup(CyclicPermutationGroup(3))
sage: chi = G.character(G.character_table()[1])
sage: G.molien_series(chi, prec=10)
t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 9*t^6 + 12*t^7 + 15*t^8 + 18*t^9 + 22*t^10 + O(t^11)
::
sage: K = GF(5)
sage: S = MatrixGroup(SymmetricGroup(4))
sage: G = MatrixGroup([matrix(K,4,4,[K(y) for u in m.list() for y in u])for m in S.gens()])
sage: G.molien_series(return_series=False)
1/(t^10 - t^9 - t^8 + 2*t^5 - t^2 - t + 1)
::
sage: i = GF(7)(3)
sage: G = MatrixGroup([[i^3,0,0,-i^3],[i^2,0,0,-i^2]])
sage: chi = G.character(G.character_table()[4])
sage: G.molien_series(chi)
3*t^5 + 6*t^11 + 9*t^17 + 12*t^23 + O(t^25)
"""
if not self.is_finite():
raise NotImplementedError("only implemented for finite groups")
if chi is None:
chi = self.trivial_character()
M = self.matrix_space()
R = FractionField(self.base_ring())
N = self.order()
if R.characteristic() == 0:
P = PolynomialRing(R, variable)
t = P.gen()
#it is possible the character is over a larger cyclotomic field
K = chi.values()[0].parent()
if K.degree() != 1:
if R.degree() != 1:
L = K.composite_fields(R)[0]
else:
L = K
else:
L = R
mol = P(0)
for g in self:
mol += L(chi(g)) / (M.identity_matrix()-t*g.matrix()).det().change_ring(L)
elif R.characteristic().divides(N):
raise NotImplementedError("characteristic cannot divide group order")
else: #char p>0
#find primitive Nth roots of unity over base ring and QQ
F = cyclotomic_polynomial(N).change_ring(R)
w = F.roots(ring=R.algebraic_closure(), multiplicities=False)[0]
#don't need to extend further in this case since the order of
#the roots of unity in the character divide the order of the group
L = CyclotomicField(N, 'v')
v = L.gen()
#construct Molien series
P = PolynomialRing(L, variable)
t = P.gen()
mol = P(0)
for g in self:
#construct Phi
phi = L(chi(g))
for e in g.matrix().eigenvalues():
#find power such that w**n = e
n = 1
while w**n != e and n < N+1:
n += 1
#raise v to that power
phi *= (1-t*v**n)
mol += P(1)/phi
#We know the coefficients will be integers
mol = mol.numerator().change_ring(ZZ) / mol.denominator().change_ring(ZZ)
#divide by group order
mol /= N
if return_series:
PS = PowerSeriesRing(ZZ, variable, default_prec=prec)
return PS(mol)
return mol
def molien_series(self,
chi=None,
return_series=True,
prec=20,
variable='t'):
r"""
Compute the Molien series of this finite group with respect to the
character ``chi``. It can be returned either as a rational function
in one variable or a power series in one variable. The base field
must be a finite field, the rationals, or a cyclotomic field.
Note that the base field characteristic cannot divide the group
order (i.e., the non-modular case).
ALGORITHM:
For a finite group `G` in characteristic zero we construct the Molien series as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\text{det}(I-tg)},
where `I` is the identity matrix and `t` an indeterminate.
For characteristic `p` not dividing the order of `G`, let `k` be the base field
and `N` the order of `G`. Define `\lambda` as a primitive `N`-th root of unity over `k`
and `\omega` as a primitive `N`-th root of unity over `\QQ`. For each `g \in G`
define `k_i(g)` to be the positive integer such that
`e_i = \lambda^{k_i(g)}` for each eigenvalue `e_i` of `g`. Then the Molien series
is computed as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\prod_{i=1}^n(1 - t\omega^{k_i(g)})},
where `t` is an indeterminant. [Dec1998]_
INPUT:
- ``chi`` -- (default: trivial character) a linear group character of this group
- ``return_series`` -- boolean (default: ``True``) if ``True``, then returns
the Molien series as a power series, ``False`` as a rational function
- ``prec`` -- integer (default: 20); power series default precision
- ``variable`` -- string (default: ``'t'``); Variable name for the Molien series
OUTPUT: single variable rational function or power series with integer coefficients
EXAMPLES::
sage: MatrixGroup(matrix(QQ,2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups
sage: MatrixGroup(matrix(GF(3),2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: characteristic cannot divide group order
Tetrahedral Group::
sage: K.<i> = CyclotomicField(4)
sage: Tetra = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [0,i, -i,0])
sage: Tetra.molien_series(prec=30)
1 + t^8 + 2*t^12 + t^16 + 2*t^20 + 3*t^24 + 2*t^28 + O(t^30)
sage: mol = Tetra.molien_series(return_series=False); mol
(t^8 - t^4 + 1)/(t^16 - t^12 - t^4 + 1)
sage: mol.parent()
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
sage: chi = Tetra.character(Tetra.character_table()[1])
sage: Tetra.molien_series(chi, prec=30, variable='u')
u^6 + u^14 + 2*u^18 + u^22 + 2*u^26 + 3*u^30 + 2*u^34 + O(u^36)
sage: chi = Tetra.character(Tetra.character_table()[2])
sage: Tetra.molien_series(chi)
t^10 + t^14 + t^18 + 2*t^22 + 2*t^26 + O(t^30)
::
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: mol = S3.molien_series(prec=10); mol
1 + t + 2*t^2 + 3*t^3 + 4*t^4 + 5*t^5 + 7*t^6 + 8*t^7 + 10*t^8 + 12*t^9 + O(t^10)
sage: mol.parent()
Power Series Ring in t over Integer Ring
Octahedral Group::
sage: K.<v> = CyclotomicField(8)
sage: a = v-v^3 #sqrt(2)
sage: i = v^2
sage: Octa = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [(1+i)/a,0, 0,(1-i)/a])
sage: Octa.molien_series(prec=30)
1 + t^8 + t^12 + t^16 + t^18 + t^20 + 2*t^24 + t^26 + t^28 + O(t^30)
Icosahedral Group::
sage: K.<v> = CyclotomicField(10)
sage: z5 = v^2
sage: i = z5^5
sage: a = 2*z5^3 + 2*z5^2 + 1 #sqrt(5)
sage: Ico = MatrixGroup([[z5^3,0, 0,z5^2], [0,1, -1,0], [(z5^4-z5)/a, (z5^2-z5^3)/a, (z5^2-z5^3)/a, -(z5^4-z5)/a]])
sage: Ico.molien_series(prec=40)
1 + t^12 + t^20 + t^24 + t^30 + t^32 + t^36 + O(t^40)
::
sage: G = MatrixGroup(CyclicPermutationGroup(3))
sage: chi = G.character(G.character_table()[1])
sage: G.molien_series(chi, prec=10)
t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 9*t^6 + 12*t^7 + 15*t^8 + 18*t^9 + 22*t^10 + O(t^11)
::
sage: K = GF(5)
sage: S = MatrixGroup(SymmetricGroup(4))
sage: G = MatrixGroup([matrix(K,4,4,[K(y) for u in m.list() for y in u])for m in S.gens()])
sage: G.molien_series(return_series=False)
1/(t^10 - t^9 - t^8 + 2*t^5 - t^2 - t + 1)
::
sage: i = GF(7)(3)
sage: G = MatrixGroup([[i^3,0,0,-i^3],[i^2,0,0,-i^2]])
sage: chi = G.character(G.character_table()[4])
sage: G.molien_series(chi)
3*t^5 + 6*t^11 + 9*t^17 + 12*t^23 + O(t^25)
"""
if not self.is_finite():
raise NotImplementedError("only implemented for finite groups")
if chi is None:
chi = self.trivial_character()
M = self.matrix_space()
R = FractionField(self.base_ring())
N = self.order()
if R.characteristic() == 0:
P = PolynomialRing(R, variable)
t = P.gen()
#it is possible the character is over a larger cyclotomic field
K = chi.values()[0].parent()
if K.degree() != 1:
if R.degree() != 1:
L = K.composite_fields(R)[0]
else:
L = K
else:
L = R
mol = P(0)
for g in self:
mol += L(chi(g)) / (M.identity_matrix() -
t * g.matrix()).det().change_ring(L)
elif R.characteristic().divides(N):
raise NotImplementedError(
"characteristic cannot divide group order")
else: #char p>0
#find primitive Nth roots of unity over base ring and QQ
F = cyclotomic_polynomial(N).change_ring(R)
w = F.roots(ring=R.algebraic_closure(), multiplicities=False)[0]
#don't need to extend further in this case since the order of
#the roots of unity in the character divide the order of the group
L = CyclotomicField(N, 'v')
v = L.gen()
#construct Molien series
P = PolynomialRing(L, variable)
t = P.gen()
mol = P(0)
for g in self:
#construct Phi
phi = L(chi(g))
for e in g.matrix().eigenvalues():
#find power such that w**n = e
n = 1
while w**n != e and n < N + 1:
n += 1
#raise v to that power
phi *= (1 - t * v**n)
mol += P(1) / phi
#We know the coefficients will be integers
mol = mol.numerator().change_ring(ZZ) / mol.denominator().change_ring(
ZZ)
#divide by group order
mol /= N
if return_series:
PS = PowerSeriesRing(ZZ, variable, default_prec=prec)
return PS(mol)
return mol
def attack(m, q, r=4, sigma=3.0, subfield_only=False):
K = CyclotomicField(m, 'z')
z = K.gen()
OK = K.ring_of_integers()
G = K.galois_group()
n = euler_phi(m)
mprime = m / r
nprime = euler_phi(mprime)
Gprime = [tau for tau in G if tau(z**r) == z**r]
R = PolynomialRing(IntegerRing(), 'a')
a = R.gen()
phim = a**n + 1
D = DiscreteGaussianDistributionIntegerSampler(sigma)
print "sampling f,g"
while True:
f = sum([D() * z**i for i in range(n)])
fx = sum([f[i] * a**i for i in range(n)])
res = inverse(fx, phim, q)
if res[0]:
f_inv = sum([res[1][i] * z**i for i in range(n)])
print "f_inv * f = %s (mod %d)" % ((f * f_inv).mod(q), q)
break
g = sum([D() * z**i for i in range(n)])
print "done sampling f, g"
#h = [g*f^{-1)]_q
h = (g * f_inv).mod(q)
lognorm_f = log(f.vector().norm(), 2)
lognorm_g = log(g.vector().norm(), 2)
print "f*h - g = %s" % (f * h - g).mod(q)
print "log q = ", log(q, 2).n(precision)
print "log |f| = %s, log |g| = %s" % (lognorm_f.n(precision),
lognorm_g.n(precision))
print "log |(f,g)| = ", log(
sqrt(f.vector().norm()**2 + g.vector().norm()**2), 2).n(precision)
print "begin computing N(f), N(g), N(h), Tr(h), fbar"
fprime = norm(f, Gprime)
gprime = norm(g, Gprime)
hprime = norm(h, Gprime).mod(q)
htr = trace(h, Gprime)
fbar = prod([tau(f) for tau in Gprime[1:]])
print "end computing N(f), N(g), N(h), Tr(h), fbar"
lognorm_fp = log(fprime.vector().norm(), 2)
lognorm_gp = log(gprime.vector().norm(), 2)
print "%d * log |f| - log |f'| = %s" % (r, r * lognorm_f.n(precision) -
lognorm_fp.n(precision))
print "log |(f', g')| = ", log(
sqrt(fprime.vector().norm()**2 + gprime.vector().norm()**2),
2).n(precision)
print "log |N(f), Tr(g fbar)| = ", log(
sqrt(fprime.vector().norm()**2 +
trace(g * fbar, Gprime).vector().norm()**2), 2).n(precision)
#(fprime, gprime) lies in the lattice \Lambda_hprime^q
print "f'*h' - g' = %s " % (hprime * fprime - gprime).mod(q)
print "N(f) Tr(h) - Tr(g fbar) = %s" % (htr * fprime -
trace(g * fbar, Gprime)).mod(q)
if not subfield_only:
ntru_full = NTRU(h, K, q)
full_sv = ntru_full.shortest_vector()
print "log |v| = %s" % log(full_sv.norm(), 2).n(precision)
ntru_subfield = NTRU_subfield(hprime, q, nprime, r)
ntru_trace_subfield = NTRU_subfield(htr, q, nprime, r)
print "begin computing Shortest Vector of subfield lattice"
norm_sv = ntru_subfield.shortest_vector()
tr_sv = ntru_trace_subfield.shortest_vector()
print "end computing Shortest Vector of subfield lattice"
norm_xp = sum(
[coerce(Integer, norm_sv[i]) * z**(r * i) for i in range(nprime)])
tr_xp = sum(
[coerce(Integer, tr_sv[i]) * z**(r * i) for i in range(nprime)])
print "Norm map: log |(x',y')| = ", log(norm_sv.norm(), 2).n(precision)
print "Trace map: log |(x', y')| = ", log(tr_sv.norm(), 2).n(precision)
#test if xprime belongs to <fprime>
mat = []
for i in range(nprime):
coordinate = (fprime * z**(r * i)).vector().list()
mat.append([coordinate[r * j] for j in range(nprime)])
FL = IntegerLattice(mat)
print norm_sv[:nprime] in FL
print tr_sv[:nprime] in FL
norm_x = norm_xp
norm_y = mod_q(norm_x * h, q)
tr_x = tr_xp
tr_y = mod_q(tr_x * h, q)
print "Norm map: log |(x,y)| = ", log(
sqrt(norm_x.vector().norm()**2 + norm_y.vector().norm()**2),
2).n(precision)
print "Trace map: log |(x,y)| = ", log(
sqrt(tr_x.vector().norm()**2 + tr_y.vector().norm()**2),
2).n(precision)
def Omega_ge(a, exponents):
r"""
Return `\Omega_{\ge}` of the expression specified by the input.
To be more precise, calculate
.. MATH::
\Omega_{\ge} \frac{\mu^a}{
(1 - z_0 \mu^{e_0}) \dots (1 - z_{n-1} \mu^{e_{n-1}})}
and return its numerator and a factorization of its denominator.
Note that `z_0`, ..., `z_{n-1}` only appear in the output, but not in the
input.
INPUT:
- ``a`` -- an integer
- ``exponents`` -- a tuple of integers
OUTPUT:
A pair representing a quotient as follows: Its first component is the
numerator as a Laurent polynomial, its second component a factorization
of the denominator as a tuple of Laurent polynomials, where each
Laurent polynomial `z` represents a factor `1 - z`.
The parents of these Laurent polynomials is always a
Laurent polynomial ring in `z_0`, ..., `z_{n-1}` over `\ZZ`, where
`n` is the length of ``exponents``.
EXAMPLES::
sage: from sage.rings.polynomial.omega import Omega_ge
sage: Omega_ge(0, (1, -2))
(1, (z0, z0^2*z1))
sage: Omega_ge(0, (1, -3))
(1, (z0, z0^3*z1))
sage: Omega_ge(0, (1, -4))
(1, (z0, z0^4*z1))
sage: Omega_ge(0, (2, -1))
(z0*z1 + 1, (z0, z0*z1^2))
sage: Omega_ge(0, (3, -1))
(z0*z1^2 + z0*z1 + 1, (z0, z0*z1^3))
sage: Omega_ge(0, (4, -1))
(z0*z1^3 + z0*z1^2 + z0*z1 + 1, (z0, z0*z1^4))
sage: Omega_ge(0, (1, 1, -2))
(-z0^2*z1*z2 - z0*z1^2*z2 + z0*z1*z2 + 1, (z0, z1, z0^2*z2, z1^2*z2))
sage: Omega_ge(0, (2, -1, -1))
(z0*z1*z2 + z0*z1 + z0*z2 + 1, (z0, z0*z1^2, z0*z2^2))
sage: Omega_ge(0, (2, 1, -1))
(-z0*z1*z2^2 - z0*z1*z2 + z0*z2 + 1, (z0, z1, z0*z2^2, z1*z2))
::
sage: Omega_ge(0, (2, -2))
(-z0*z1 + 1, (z0, z0*z1, z0*z1))
sage: Omega_ge(0, (2, -3))
(z0^2*z1 + 1, (z0, z0^3*z1^2))
sage: Omega_ge(0, (3, 1, -3))
(-z0^3*z1^3*z2^3 + 2*z0^2*z1^3*z2^2 - z0*z1^3*z2
+ z0^2*z2^2 - 2*z0*z2 + 1,
(z0, z1, z0*z2, z0*z2, z0*z2, z1^3*z2))
::
sage: Omega_ge(0, (3, 6, -1))
(-z0*z1*z2^8 - z0*z1*z2^7 - z0*z1*z2^6 - z0*z1*z2^5 - z0*z1*z2^4 +
z1*z2^5 - z0*z1*z2^3 + z1*z2^4 - z0*z1*z2^2 + z1*z2^3 -
z0*z1*z2 + z0*z2^2 + z1*z2^2 + z0*z2 + z1*z2 + 1,
(z0, z1, z0*z2^3, z1*z2^6))
TESTS::
sage: Omega_ge(0, (2, 2, 1, 1, 1, -1, -1))[0].number_of_terms() # long time
1695
sage: Omega_ge(0, (2, 2, 1, 1, 1, 1, 1, -1, -1))[0].number_of_terms() # not tested (too long, 1 min)
27837
::
sage: Omega_ge(1, (2,))
(1, (z0,))
"""
import logging
logger = logging.getLogger(__name__)
logger.info('Omega_ge: a=%s, exponents=%s', a, exponents)
from sage.arith.all import lcm, srange
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
from sage.rings.number_field.number_field import CyclotomicField
if not exponents or any(e == 0 for e in exponents):
raise NotImplementedError
rou = sorted(set(abs(e) for e in exponents) - set([1]))
ellcm = lcm(rou)
B = CyclotomicField(ellcm, 'zeta')
zeta = B.gen()
z_names = tuple('z{}'.format(i) for i in range(len(exponents)))
L = LaurentPolynomialRing(B, ('t', ) + z_names, len(z_names) + 1)
t = L.gens()[0]
Z = LaurentPolynomialRing(ZZ, z_names, len(z_names))
powers = {i: L(zeta**(ellcm // i)) for i in rou}
powers[2] = L(-1)
powers[1] = L(1)
exponents_and_values = tuple(
(e, tuple(powers[abs(e)]**j * z for j in srange(abs(e))))
for z, e in zip(L.gens()[1:], exponents))
x = tuple(v for e, v in exponents_and_values if e > 0)
y = tuple(v for e, v in exponents_and_values if e < 0)
def subs_power(expression, var, exponent):
r"""
Substitute ``var^exponent`` by ``var`` in ``expression``.
It is assumed that ``var`` only occurs with exponents
divisible by ``exponent``.
"""
p = tuple(var.dict().popitem()[0]).index(
1) # var is the p-th generator
def subs_e(e):
e = list(e)
assert e[p] % exponent == 0
e[p] = e[p] // exponent
return tuple(e)
parent = expression.parent()
result = parent(
{subs_e(e): c
for e, c in iteritems(expression.dict())})
return result
def de_power(expression):
expression = Z(expression)
for e, var in zip(exponents, Z.gens()):
if abs(e) == 1:
continue
expression = subs_power(expression, var, abs(e))
return expression
logger.debug('Omega_ge: preparing denominator')
factors_denominator = tuple(
de_power(1 - factor) for factor in _Omega_factors_denominator_(x, y))
logger.debug('Omega_ge: preparing numerator')
numerator = de_power(_Omega_numerator_(a, x, y, t))
logger.info('Omega_ge: completed')
return numerator, factors_denominator
def Omega_ge(a, exponents):
r"""
Return `\Omega_{\ge}` of the expression specified by the input.
To be more precise, calculate
.. MATH::
\Omega_{\ge} \frac{\mu^a}{
(1 - z_0 \mu^{e_0}) \dots (1 - z_{n-1} \mu^{e_{n-1}})}
and return its numerator and a factorization of its denominator.
Note that `z_0`, ..., `z_{n-1}` only appear in the output, but not in the
input.
INPUT:
- ``a`` -- an integer
- ``exponents`` -- a tuple of integers
OUTPUT:
A pair representing a quotient as follows: Its first component is the
numerator as a Laurent polynomial, its second component a factorization
of the denominator as a tuple of Laurent polynomials, where each
Laurent polynomial `z` represents a factor `1 - z`.
The parents of these Laurent polynomials is always a
Laurent polynomial ring in `z_0`, ..., `z_{n-1}` over `\ZZ`, where
`n` is the length of ``exponents``.
EXAMPLES::
sage: from sage.rings.polynomial.omega import Omega_ge
sage: Omega_ge(0, (1, -2))
(1, (z0, z0^2*z1))
sage: Omega_ge(0, (1, -3))
(1, (z0, z0^3*z1))
sage: Omega_ge(0, (1, -4))
(1, (z0, z0^4*z1))
sage: Omega_ge(0, (2, -1))
(z0*z1 + 1, (z0, z0*z1^2))
sage: Omega_ge(0, (3, -1))
(z0*z1^2 + z0*z1 + 1, (z0, z0*z1^3))
sage: Omega_ge(0, (4, -1))
(z0*z1^3 + z0*z1^2 + z0*z1 + 1, (z0, z0*z1^4))
sage: Omega_ge(0, (1, 1, -2))
(-z0^2*z1*z2 - z0*z1^2*z2 + z0*z1*z2 + 1, (z0, z1, z0^2*z2, z1^2*z2))
sage: Omega_ge(0, (2, -1, -1))
(z0*z1*z2 + z0*z1 + z0*z2 + 1, (z0, z0*z1^2, z0*z2^2))
sage: Omega_ge(0, (2, 1, -1))
(-z0*z1*z2^2 - z0*z1*z2 + z0*z2 + 1, (z0, z1, z0*z2^2, z1*z2))
::
sage: Omega_ge(0, (2, -2))
(-z0*z1 + 1, (z0, z0*z1, z0*z1))
sage: Omega_ge(0, (2, -3))
(z0^2*z1 + 1, (z0, z0^3*z1^2))
sage: Omega_ge(0, (3, 1, -3))
(-z0^3*z1^3*z2^3 + 2*z0^2*z1^3*z2^2 - z0*z1^3*z2
+ z0^2*z2^2 - 2*z0*z2 + 1,
(z0, z1, z0*z2, z0*z2, z0*z2, z1^3*z2))
::
sage: Omega_ge(0, (3, 6, -1))
(-z0*z1*z2^8 - z0*z1*z2^7 - z0*z1*z2^6 - z0*z1*z2^5 - z0*z1*z2^4 +
z1*z2^5 - z0*z1*z2^3 + z1*z2^4 - z0*z1*z2^2 + z1*z2^3 -
z0*z1*z2 + z0*z2^2 + z1*z2^2 + z0*z2 + z1*z2 + 1,
(z0, z1, z0*z2^3, z1*z2^6))
TESTS::
sage: Omega_ge(0, (2, 2, 1, 1, 1, 1, 1, -1, -1))[0].number_of_terms() # long time
27837
::
sage: Omega_ge(1, (2,))
(1, (z0,))
"""
import logging
logger = logging.getLogger(__name__)
logger.info('Omega_ge: a=%s, exponents=%s', a, exponents)
from sage.arith.all import lcm, srange
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
from sage.rings.number_field.number_field import CyclotomicField
if not exponents or any(e == 0 for e in exponents):
raise NotImplementedError
rou = sorted(set(abs(e) for e in exponents) - set([1]))
ellcm = lcm(rou)
B = CyclotomicField(ellcm, 'zeta')
zeta = B.gen()
z_names = tuple('z{}'.format(i) for i in range(len(exponents)))
L = LaurentPolynomialRing(B, ('t',) + z_names, len(z_names) + 1)
t = L.gens()[0]
Z = LaurentPolynomialRing(ZZ, z_names, len(z_names))
powers = {i: L(zeta**(ellcm//i)) for i in rou}
powers[2] = L(-1)
powers[1] = L(1)
exponents_and_values = tuple(
(e, tuple(powers[abs(e)]**j * z for j in srange(abs(e))))
for z, e in zip(L.gens()[1:], exponents))
x = tuple(v for e, v in exponents_and_values if e > 0)
y = tuple(v for e, v in exponents_and_values if e < 0)
def subs_power(expression, var, exponent):
r"""
Substitute ``var^exponent`` by ``var`` in ``expression``.
It is assumed that ``var`` only occurs with exponents
divisible by ``exponent``.
"""
p = tuple(var.dict().popitem()[0]).index(1) # var is the p-th generator
def subs_e(e):
e = list(e)
assert e[p] % exponent == 0
e[p] = e[p] // exponent
return tuple(e)
parent = expression.parent()
result = parent({subs_e(e): c for e, c in iteritems(expression.dict())})
return result
def de_power(expression):
expression = Z(expression)
for e, var in zip(exponents, Z.gens()):
if abs(e) == 1:
continue
expression = subs_power(expression, var, abs(e))
return expression
logger.debug('Omega_ge: preparing denominator')
factors_denominator = tuple(de_power(1 - factor)
for factor in _Omega_factors_denominator_(x, y))
logger.debug('Omega_ge: preparing numerator')
numerator = de_power(_Omega_numerator_(a, x, y, t))
logger.info('Omega_ge: completed')
return numerator, factors_denominator