Ejemplo n.º 1
0
    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
Ejemplo n.º 2
0
    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
Ejemplo n.º 3
0
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())
Ejemplo n.º 4
0
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())