Python PowerSeriesRing.gen方法代码示例

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]
    def local_coordinates_at_nonweierstrass(self, P, prec=20, name='t'):
        """
        For a non-Weierstrass point `P = (a,b)` on the hyperelliptic
        curve `y^2 = f(x)`, return `(x(t), y(t))` such that `(y(t))^2 = f(x(t))`,
        where `t = x - a` is the local parameter.

        INPUT:

        - ``P = (a, b)`` -- a non-Weierstrass point on self
        - ``prec`` --  desired precision of the local coordinates
        - ``name`` -- gen of the power series ring (default: ``t``)

        OUTPUT:

        `(x(t),y(t))` such that `y(t)^2 = f(x(t))` and `t = x - a`
        is the local parameter at `P`

        EXAMPLES::

            sage: R.<x> = QQ['x']
            sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
            sage: P = H(1,6)
            sage: x,y = H.local_coordinates_at_nonweierstrass(P,prec=5)
            sage: x
            1 + t + O(t^5)
            sage: y
            6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5)
            sage: Q = H(-2,12)
            sage: x,y = H.local_coordinates_at_nonweierstrass(Q,prec=5)
            sage: x
            -2 + t + O(t^5)
            sage: y
            12 - 19/2*t - 19/32*t^2 + 61/256*t^3 - 5965/24576*t^4 + O(t^5)

        AUTHOR:

            - Jennifer Balakrishnan (2007-12)
        """
        d = P[1]
        if d == 0:
            raise TypeError("P = %s is a Weierstrass point. Use local_coordinates_at_weierstrass instead!"%P)
        pol = self.hyperelliptic_polynomials()[0]
        L = PowerSeriesRing(self.base_ring(), name)
        t = L.gen()
        L.set_default_prec(prec)
        K = PowerSeriesRing(L, 'x')
        pol = K(pol)
        x = K.gen()
        b = P[0]
        f = pol(t+b)
        for i in range((RR(log(prec)/log(2))).ceil()):
            d = (d + f/d)/2
        return t+b+O(t**(prec)), d + O(t**(prec))

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]
 def _e_bounds(self, n, prec):
     p = self._p
     prec = max(2,prec)
     R = PowerSeriesRing(ZZ,'T',prec+1)
     T = R(R.gen(),prec +1)
     w = (1+T)**(p**n) - 1
     return [infinity] + [valuation(w[j],p) for j in range(1,min(w.degree()+1,prec))]

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]
    def local_coordinates_at_weierstrass(self, P, prec=20, name="t"):
        """                                                                                                 
        For a finite Weierstrass point on the hyperelliptic                                
        curve y^2 = f(x), returns (x(t), y(t)) such that                                   
        (y(t))^2 = f(x(t)), where t = y is the local parameter.

        INPUT:
            - P a finite Weierstrass point on self
            - prec: desired precision of the local coordinates
            - name: gen of the power series ring (default: 't')

        OUTPUT:

        (x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
        is the local parameter at P                                             
                                                                                                            
        EXAMPLES:                                                                                           
            sage: R.<x> = QQ['x']                                                              
            sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)               
            sage: A = H(4, 0)

            sage: x, y = H.local_coordinates_at_weierstrass(A, prec=7)

            sage: x
            4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
            sage: y 
            t + O(t^7)
            sage: B = H(-5, 0)
            sage: x, y = H.local_coordinates_at_weierstrass(B, prec=5)
            sage: x
            -5 + 1/1260*t^2 + 887/2000376000*t^4 + O(t^5)
            sage: y
            t + O(t^5)
                                                                                                            
        AUTHOR:                                                                                             
            - Jennifer Balakrishnan (2007-12)

            - Francis Clarke (2012-08-26)
        """
        if P[1] != 0:
            raise TypeError, "P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!" % P
        L = PowerSeriesRing(self.base_ring(), name)
        t = L.gen()
        pol = self.hyperelliptic_polynomials()[0]
        pol_prime = pol.derivative()
        b = P[0]
        t2 = t ** 2
        c = b + t2 / pol_prime(b)
        c = c.add_bigoh(prec)
        for _ in range(1 + log(prec, 2)):
            c -= (pol(c) - t2) / pol_prime(c)
        return (c, t.add_bigoh(prec))

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]
    def zeta_series(self, n, t):
        """
        Return the zeta series.

        Compute a power series approximation to the zeta function of a
        scheme over a finite field.

        INPUT:

        -  ``n`` -- the number of terms of the power series to
           compute

        -  ``t`` -- the variable which the series should be
           returned


        OUTPUT:

        A power series approximating the zeta function of self

        EXAMPLES::

            sage: P.<x> = PolynomialRing(GF(3))
            sage: C = HyperellipticCurve(x^3+x^2+1)
            sage: R.<t> = PowerSeriesRing(Integers())
            sage: C.zeta_series(4,t)
            1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
            sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
            1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)

        Note that this function depends on count_points, which is only
        defined for prime order fields::

            sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
            Traceback (most recent call last):
            ...
            NotImplementedError: Point counting only implemented for schemes over prime fields
        """

        F = self.base_ring()
        if not F.is_finite():
            raise TypeError('zeta functions only defined for schemes over finite fields')
        try:
            a = self.count_points(n)
        except AttributeError:
            raise NotImplementedError('count_points() required but not implemented')
        R = PowerSeriesRing(Rationals(), 'u')
        u = R.gen()
        temp = sum(a[i-1]*(u.O(n+1))**i/i for i in range(1,n+1))
        temp2 = temp.exp()
        return(temp2(t).O(n+1))

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]

#.........这里部分代码省略.........
            sage: MFSeriesConstructor(prec=3).J_inv_ZZ()
            q^-1 + 31/72 + 1823/27648*q + O(q^2)
            sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
            q^-1 + 79/200 + 42877/640000*q + O(q^2)
            sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
            Laurent Series Ring in q over Rational Field

            sage: MFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
            q^-1 + 3/8 + 69/1024*q + O(q^2)
        """

        F1       = lambda a,b:   self._series_ring(
                       [ ZZ(0) ]
                       + [
                           rising_factorial(a,k) * rising_factorial(b,k) / (ZZ(k).factorial())**2
                           * sum(ZZ(1)/(a+j) + ZZ(1)/(b+j) - ZZ(2)/ZZ(1+j)
                                  for j in range(ZZ(0),ZZ(k))
                             )
                           for k in range(ZZ(1), ZZ(self._prec+1))
                       ],
                       ZZ(self._prec+1)
                   )

        F        = lambda a,b,c: self._series_ring(
                       [
                         rising_factorial(a,k) * rising_factorial(b,k) / rising_factorial(c,k) / ZZ(k).factorial()
                         for k in range(ZZ(0), ZZ(self._prec+1))
                       ],
                       ZZ(self._prec+1)
                   )
        a        = self._group.alpha()
        b        = self._group.beta()
        Phi      = F1(a,b) / F(a,b,ZZ(1))
        q        = self._series_ring.gen()

        # the current implementation of power series reversion is slow
        # J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reverse())

        temp_f   = (q*Phi.exp()).polynomial()
        new_f    = temp_f.revert_series(temp_f.degree()+1)
        J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree()+1)))

        return J_inv_ZZ

    @cached_method
    def f_rho_ZZ(self):
        r"""
        Return the rational Fourier expansion of ``f_rho``,
        where the parameter ``d`` is replaced by ``1``.

        .. NOTE:

        The Fourier expansion of ``f_rho`` for ``d!=1``
        is given by ``f_rho_ZZ(q/d)``.

        EXAMPLES::

            sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
            sage: MFSeriesConstructor(prec=3).f_rho_ZZ()
            1 + 5/36*q + 5/6912*q^2 + O(q^3)
            sage: MFSeriesConstructor(group=5, prec=3).f_rho_ZZ()
            1 + 7/100*q + 21/160000*q^2 + O(q^3)
            sage: MFSeriesConstructor(group=5, prec=3).f_rho_ZZ().parent()
            Power Series Ring in q over Rational Field

            sage: MFSeriesConstructor(group=infinity, prec=3).f_rho_ZZ()

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]

#.........这里部分代码省略.........
            Hecke triangle group for n = 3
            sage: MFC.prec()
            10
            sage: MFC.d().parent()
            Fraction Field of Univariate Polynomial Ring in d over Integer Ring
            sage: MFC._ZZseries_ring
            Power Series Ring in q over Rational Field

            sage: MFSeriesConstructor(set_d=CC(1))
            Power series constructor for Hecke modular forms for n=3, base ring=Complex Field with 53 bits of precision
            with (basic series) precision 10 with parameter d=1.00000000000000
            
            sage: MFSeriesConstructor(group=4, fix_d=True)
            Power series constructor for Hecke modular forms for n=4, base ring=Rational Field
            with (basic series) precision 10 with parameter d=1/256

            sage: MFSeriesConstructor(group=5, fix_d=True)
            Power series constructor for Hecke modular forms for n=5, base ring=Real Field with 53 bits of precision
            with (basic series) precision 10 with parameter d=0.00705223418128563
        """

        self._group          = group
        self._base_ring      = base_ring
        self._prec           = prec
        self._fix_d          = fix_d
        self._set_d          = set_d
        self._d_num_prec     = d_num_prec

        if (set_d):
            self._coeff_ring = FractionField(base_ring)
            self._d          = set_d
        else:
            self._coeff_ring = FractionField(PolynomialRing(base_ring,"d"))
            self._d          = self._coeff_ring.gen()

        self._ZZseries_ring  = PowerSeriesRing(QQ,'q',default_prec=self._prec)
        self._qseries_ring   = PowerSeriesRing(self._coeff_ring,'q',default_prec=self._prec)

    def _repr_(self):
        r"""
        Return the string representation of ``self``.

        EXAMPLES::

            sage: MFSeriesConstructor(group=4, fix_d=True)
            Power series constructor for Hecke modular forms for n=4, base ring=Rational Field
            with (basic series) precision 10 with parameter d=1/256

            sage: MFSeriesConstructor(group=5)
            Power series constructor for Hecke modular forms for n=5, base ring=Integer Ring
            with (basic series) precision 10 with formal parameter d
        """

        if (self._set_d):
            return "Power series constructor for Hecke modular forms for n={}, base ring={} with (basic series) precision {} with parameter d={}".\
                format(self._group.n(), self._base_ring, self._prec, self._d)
        else:
            return "Power series constructor for Hecke modular forms for n={}, base ring={} with (basic series) precision {} with formal parameter d".\
                format(self._group.n(), self._base_ring, self._prec)

    def group(self):
        r"""
        Return the (Hecke triangle) group of ``self``.

        EXAMPLES::

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]

#.........这里部分代码省略.........
            sage: JFSeriesConstructor(prec=3).J_inv_ZZ()
            q^-1 + 31/72 + 1823/27648*q + O(q^2)
            sage: JFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
            q^-1 + 79/200 + 42877/640000*q + O(q^2)
            sage: JFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
            Laurent Series Ring in q over Rational Field

            sage: JFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
            q^-1 + 3/8 + 69/1024*q + O(q^2)
        """

        F1       = lambda a,b:   self._qseries_ring(
                       [ ZZ(0) ]
                       + [
                           rising_factorial(a,k) * rising_factorial(b,k) / (ZZ(k).factorial())**2
                           * sum(ZZ(1)/(a+j) + ZZ(1)/(b+j) - ZZ(2)/ZZ(1+j)
                                  for j in range(ZZ(0),ZZ(k))
                             )
                           for k in range(ZZ(1), ZZ(self._prec+1))
                       ],
                       ZZ(self._prec+1)
                   )

        F        = lambda a,b,c: self._qseries_ring(
                       [
                         rising_factorial(a,k) * rising_factorial(b,k) / rising_factorial(c,k) / ZZ(k).factorial()
                         for k in range(ZZ(0), ZZ(self._prec+1))
                       ],
                       ZZ(self._prec+1)
                   )
        a        = self._group.alpha()
        b        = self._group.beta()
        Phi      = F1(a,b) / F(a,b,ZZ(1))
        q        = self._qseries_ring.gen()

        # the current implementation of power series reversion is slow
        # J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reversion())

        temp_f   = (q*Phi.exp()).polynomial()
        new_f    = temp_f.revert_series(temp_f.degree()+1)
        J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree()+1)))
        q        = self._series_ring.gen()
        J_inv_ZZ = sum([J_inv_ZZ.coefficients()[m] * q**J_inv_ZZ.exponents()[m] for m in range(len(J_inv_ZZ.coefficients()))]) + O(q**J_inv_ZZ.prec())

        return J_inv_ZZ

    @cached_method
    def f_rho_ZZ(self):
        r"""
        Return the rational Fourier expansion of ``f_rho``,
        where the parameter ``d`` is replaced by ``1``.

        .. NOTE:

        The Fourier expansion of ``f_rho`` for ``d!=1``
        is given by ``f_rho_ZZ(q/d)``.

        EXAMPLES::

            sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
            sage: JFSeriesConstructor(prec=3).f_rho_ZZ()
            1 + 5/36*q + 5/6912*q^2 + O(q^3)
            sage: JFSeriesConstructor(group=5, prec=3).f_rho_ZZ()
            1 + 7/100*q + 21/160000*q^2 + O(q^3)
            sage: JFSeriesConstructor(group=5, prec=3).f_rho_ZZ().parent()
            Power Series Ring in q over Rational Field

# 需要导入模块: from sage.rings.all import PowerSeriesRing [as 别名]
# 或者: from sage.rings.all.PowerSeriesRing import gen [as 别名]
    def series(self, n=2, quadratic_twist=+1, prec=5):
        r"""
        Returns the `n`-th approximation to the `p`-adic L-series as a
        power series in `T` (corresponding to `\gamma-1` with
        `\gamma=1+p` as a generator of `1+p\ZZ_p`).  Each coefficient
        is a `p`-adic number whose precision is provably correct.
        
        Here the normalization of the `p`-adic L-series is chosen such
        that `L_p(J,1) = (1-1/\alpha)^2 L(J,1)/\Omega_J` where
        `\alpha` is the unit root

        INPUT:
        
            - ``n`` - (default: 2) a positive integer
            - ``quadratic_twist`` - (default: +1) a fundamental
              discriminant of a quadratic field, coprime to the
              conductor of the curve
            - ``prec`` - (default: 5) maximal number of terms of the
              series to compute; to compute as many as possible just
              give a very large number for ``prec``; the result will
              still be correct.

        ALIAS: power_series is identical to series.

        EXAMPLES:

	    sage: J = J0(188)[0]
	    sage: p = 7
	    sage: L = J.padic_lseries(p)
	    sage: L.is_ordinary()
	    True
	    sage: f = L.series(2)
	    sage: f[0]
	    O(7^20)
	    sage: f[1].norm()
	    3 + 4*7 + 3*7^2 + 6*7^3 + 5*7^4 + 5*7^5 + 6*7^6 + 4*7^7 + 5*7^8 + 7^10 + 5*7^11 + 4*7^13 + 4*7^14 + 5*7^15 + 2*7^16 + 5*7^17 + 7^18 + 7^19 + O(7^20)

        """
        n = ZZ(n)
        if n < 1:
            raise ValueError, "n (=%s) must be a positive integer"%n
        if not self.is_ordinary():
            raise ValueError, "p (=%s) must be an ordinary prime"%p
        # check if the conditions on quadratic_twist are satisfied
        D = ZZ(quadratic_twist)
        if D != 1:
            if D % 4 == 0:
                d = D//4
                if not d.is_squarefree() or d % 4 == 1:
                    raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
            else:
                if not D.is_squarefree() or D % 4 != 1:
                    raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
            if gcd(D,self._p) != 1:
                raise ValueError, "quadratic twist (=%s) must be coprime to p (=%s) "%(D,self._p)
            if gcd(D,self._E.conductor())!= 1:
                for ell in prime_divisors(D):
                    if valuation(self._E.conductor(),ell) > valuation(D,ell) :
                        raise ValueError, "can not twist a curve of conductor (=%s) by the quadratic twist (=%s)."%(self._E.conductor(),D)
                    
            
        p = self._p
        if p == 2 and self._normalize :
            print 'Warning : For p=2 the normalization might not be correct !'
        #verbose("computing L-series for p=%s, n=%s, and prec=%s"%(p,n,prec))
        
#        bounds = self._prec_bounds(n,prec)
#        padic_prec = max(bounds[1:]) + 5
        padic_prec = 10
#        verbose("using p-adic precision of %s"%padic_prec)
        
        res_series_prec = min(p**(n-1), prec)
        verbose("using series precision of %s"%res_series_prec)
        
        ans = self._get_series_from_cache(n, res_series_prec,D)
        if not ans is None:
            verbose("found series in cache")
            return ans
 
        K = QQ
        gamma = K(1 + p)
        R = PowerSeriesRing(K,'T',res_series_prec)
        T = R(R.gen(),res_series_prec )
        #L = R(0) 
        one_plus_T_factor = R(1) 
        gamma_power = K(1)
        teich = self.teichmuller(padic_prec)
        p_power = p**(n-1)
#        F = Qp(p,padic_prec)

        verbose("Now iterating over %s summands"%((p-1)*p_power))
        verbose_level = get_verbose()
        count_verb = 0
        alphas = self.alpha()
        #print len(alphas)
        Lprod = []
        self._emb = 0
        if len(alphas) == 2:
            split = True
        else:
#.........这里部分代码省略.........