def _cis_iterator(self, base_ring):
r"""
The cycle index series of the species of cyclic permutations is
given by
.. math::
-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)
which is equal to
.. math::
\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}
.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/3*p[1, 1, 1] + 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
"""
from sage.combinat.sf.sf import SymmetricFunctions
p = SymmetricFunctions(base_ring).power()
zero = base_ring(0)
yield zero
for n in _integers_from(1):
res = zero
for k in divisors(n):
res += euler_phi(k) * p([k]) ** (n // k)
res /= n
yield self._weight * res
def _cis_iterator(self, base_ring):
r"""
The cycle index series of the species of cyclic permutations is
given by
.. math::
-\sum_{k=1}^\infty \phi(k)/k * log(1 - x_k)
which is equal to
.. math::
\sum_{n=1}^\infty \frac{1}{n} * \sum_{k|n} \phi(k) * x_k^{n/k}
.
EXAMPLES::
sage: P = species.CycleSpecies()
sage: cis = P.cycle_index_series()
sage: cis.coefficients(7)
[0,
p[1],
1/2*p[1, 1] + 1/2*p[2],
1/3*p[1, 1, 1] + 2/3*p[3],
1/4*p[1, 1, 1, 1] + 1/4*p[2, 2] + 1/2*p[4],
1/5*p[1, 1, 1, 1, 1] + 4/5*p[5],
1/6*p[1, 1, 1, 1, 1, 1] + 1/6*p[2, 2, 2] + 1/3*p[3, 3] + 1/3*p[6]]
"""
from sage.combinat.sf.all import SFAPower
p = SFAPower(base_ring)
zero = base_ring(0)
yield zero
for n in _integers_from(1):
res = zero
for k in divisors(n):
res += euler_phi(k)*p([k])**(n//k)
res /= n
yield self._weight*res
def cyclotomic_restriction(L,K):
r"""
Given two cyclotomic fields L and K, compute the compositum
M of K and L, and return a function and the index [M:K]. The
function is a map that acts as follows (here `M = Q(\zeta_m)`):
INPUT:
element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(\zeta_m) = \alpha`,
where we view alpha as living in `M`. (Note that `\zeta_m`
generates `M`, not `L`.)
EXAMPLES::
sage: L = CyclotomicField(12) ; N = CyclotomicField(33) ; M = CyclotomicField(132)
sage: z, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N)
sage: n
2
sage: z(L.0)
-zeta33^19*x
sage: z(L.0)(M.0)
zeta132^11
sage: z(L.0^3-L.0+1)
(zeta33^19 + zeta33^8)*x + 1
sage: z(L.0^3-L.0+1)(M.0)
zeta132^33 - zeta132^11 + 1
sage: z(L.0^3-L.0+1)(M.0) - M(L.0^3-L.0+1)
0
"""
if not L.has_coerce_map_from(K):
M = CyclotomicField(lcm(L.zeta_order(), K.zeta_order()))
f = cyclotomic_restriction_tower(M,K)
def g(x):
"""
Function returned by cyclotomic restriction.
INPUT:
element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(\zeta_m) = \alpha`,
where we view alpha as living in `M`. (Note that `\zeta_m`
generates `M`, not `L`.)
EXAMPLES::
sage: L = CyclotomicField(12)
sage: N = CyclotomicField(33)
sage: g, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N)
sage: g(L.0)
-zeta33^19*x
"""
return f(M(x))
return g, euler_phi(M.zeta_order())//euler_phi(K.zeta_order())
else:
return cyclotomic_restriction_tower(L,K), \
euler_phi(L.zeta_order())//euler_phi(K.zeta_order())
def cyclotomic_restriction(L, K):
r"""
Given two cyclotomic fields L and K, compute the compositum
M of K and L, and return a function and the index [M:K]. The
function is a map that acts as follows (here `M = Q(\zeta_m)`):
INPUT:
element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(\zeta_m) = \alpha`,
where we view alpha as living in `M`. (Note that `\zeta_m`
generates `M`, not `L`.)
EXAMPLES::
sage: L = CyclotomicField(12) ; N = CyclotomicField(33) ; M = CyclotomicField(132)
sage: z, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N)
sage: n
2
sage: z(L.0)
-zeta33^19*x
sage: z(L.0)(M.0)
zeta132^11
sage: z(L.0^3-L.0+1)
(zeta33^19 + zeta33^8)*x + 1
sage: z(L.0^3-L.0+1)(M.0)
zeta132^33 - zeta132^11 + 1
sage: z(L.0^3-L.0+1)(M.0) - M(L.0^3-L.0+1)
0
"""
if not L.has_coerce_map_from(K):
M = CyclotomicField(lcm(L.zeta_order(), K.zeta_order()))
f = cyclotomic_restriction_tower(M, K)
def g(x):
"""
Function returned by cyclotomic restriction.
INPUT:
element alpha in L
OUTPUT:
a polynomial `f(x)` in `K[x]` such that `f(\zeta_m) = \alpha`,
where we view alpha as living in `M`. (Note that `\zeta_m`
generates `M`, not `L`.)
EXAMPLES::
sage: L = CyclotomicField(12)
sage: N = CyclotomicField(33)
sage: g, n = sage.modular.modform.eisenstein_submodule.cyclotomic_restriction(L,N)
sage: g(L.0)
-zeta33^19*x
"""
return f(M(x))
return g, euler_phi(M.zeta_order()) // euler_phi(K.zeta_order())
else:
return cyclotomic_restriction_tower(L,K), \
euler_phi(L.zeta_order())//euler_phi(K.zeta_order())