Python PolynomialRing.ideal方法代码示例

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def generator_relations(self, K) :
        """
        An ideal `I` in the attach polynomial ring `R`, such that the ring of
        modular forms is a quotient of `R / I`. This ideal must be unique for `K`.
        
        INPUT:
            - `K`      -- A ring or module; The ring of Fourier coefficients.
        
        OUTPUT:
            An ideal in a polynomial ring.
            
        TESTS::
            sage: from hermitianmodularforms import *
            sage: HermitianModularFormD2_Gamma(-3).generator_relations(QQ)
            Ideal (0) of Multivariate Polynomial Ring in HE4, HE6, HE10, HE12, Hphi9 over Rational Field
            sage: HermitianModularFormD2_Gamma(-3).generator_relations(GF(2))
            Traceback (most recent call last):
            ...
            NotImplementedError: Only Fourier coefficients in a number fields are implemented.
        """
        if self.__D == -3 :
            if K is QQ or K in NumberFields() :
                R = PolynomialRing(K, self._generator_names(K))
                return R.ideal(0)

            raise NotImplementedError( "Only Fourier coefficients in a number fields are implemented." )
        
        raise NotImplementedError( "Discriminant %s is not implemented." % (self.__D,) )

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def genus(self):
        """
        Return the genus of this function field
        For now, the genus is computed using singular

        EXAMPLES::

            sage: K.<x> = FunctionField(QQ); R.<y> = K[]
            sage: L.<y> = K.extension(y^3 - (x^3 + 2*x*y + 1/x))
            sage: L.genus()
            3
        """
        # unfortunately singular can not compute the genus with the polynomial_ring()._singular_
        # object because genus method only accepts a ring of transdental degree 2 over a prime field
        # not a ring of transdental degree 1 over a rational function field of one variable

        if is_RationalFunctionField(self._base_field) and self._base_field.constant_field().is_prime_field():

            #Making the auxiliary ring which only has polynomials with integral coefficients.
            tmpAuxRing = PolynomialRing(self._base_field.constant_field(), str(self._base_field.gen())+','+str(self._ring.gen()))
            intMinPoly, d = self._make_monic_integral(self._polynomial)
            curveIdeal = tmpAuxRing.ideal(intMinPoly)

            singular.lib('normal.lib') #loading genus method in singular
            return int(curveIdeal._singular_().genus())

        else:
            raise NotImplementedError("Computation of genus over this rational function field not implemented yet")

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
def _eta_relations_helper(eta1, eta2, degree, qexp_terms, labels, verbose):
    r"""
    Helper function used by eta_poly_relations. Finds a basis for the
    space of linear relations between the first qexp_terms of the
    `q`-expansions of the monomials
    `\eta_1^i * \eta_2^j` for `0 \le i,j < degree`,
    and calculates a Groebner basis for the ideal generated by these
    relations.

    Liable to return meaningless results if qexp_terms isn't at least
    `1 + d*(m_1,m_2)` where

    .. math::

       m_i = min(0, {\text degree of the pole of $\eta_i$ at $\infty$})

    as then 1 will be in the ideal.

    EXAMPLE::

        sage: from sage.modular.etaproducts import _eta_relations_helper
        sage: r,s = EtaGroup(4).basis()
        sage: _eta_relations_helper(r,s,4,100,['a','b'],False)
        [a*b - a + 16]
        sage: _eta_relations_helper(EtaProduct(26, {2:2,13:2,26:-2,1:-2}),EtaProduct(26, {2:4,13:2,26:-4,1:-2}),3,12,['a','b'],False) # not enough terms, will return rubbish
        [1]
    """

    indices = [(i,j) for j in range(degree) for i in range(degree)]
    inf = CuspFamily(eta1.level(), 1)

    pole_at_infinity = -(min([0, eta1.order_at_cusp(inf)]) + min([0,eta2.order_at_cusp(inf)]))*degree
    if verbose: print "Trying all coefficients from q^%s to q^%s inclusive" % (-pole_at_infinity, -pole_at_infinity + qexp_terms - 1)

    rows = []
    for j in xrange(qexp_terms):
        rows.append([])
    for i in indices:
        func = (eta1**i[0]*eta2**i[1]).qexp(qexp_terms)
        for j in xrange(qexp_terms):
            rows[j].append(func[j - pole_at_infinity])
    M = matrix(rows)
    V = M.right_kernel()
    if V.dimension() == 0:
        if verbose: print "No polynomial relation of order %s valid for %s terms" % (degree, qexp_terms)
        return None
    if V.dimension() >= 1:
        #print "Found relation: "
        R = PolynomialRing(QQ, 2, labels)
        x,y = R.gens()
        relations = []
        for c in V.basis():
            relations.append(sum( [ c[v] * x**indices[v][0] * y**indices[v][1] for v in xrange(len(indices))]))
            #print relations[-1], " = 0"
        id = R.ideal(relations)
        return id.groebner_basis()

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
 def generator_relations(self, K) :
     r"""
     An ideal `I` in a polynomial ring `R`, such that the associated ring
     is `R / I`. This ideal must be unique for `K`. 
     """
     if K is QQ or K in NumberFields() :
         R = PolynomialRing(K, self._generator_names(K))
         return R.ideal(0)
         
     raise NotImplementedError

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
 def generator_relations(self, K) :
     r"""
     An ideal I in a polynomial ring R, such that the associated module
     is (R / I)_1. 
     """
     if K is QQ or K in NumberFields() :
         R = PolynomialRing(K, self._generator_names(K))
         return R.ideal(0)
         
     raise NotImplementedError

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def is_morphism(self):
        r"""
        returns ``True`` if self is a morphism (no common zero of defining polynomials).
        The map is a morphism if and only if the ideal generated by the defining polynomials
        is the unit ideal.


        OUTPUT:

        - Boolean

        EXAMPLES::

            sage: P.<x,y>=ProjectiveSpace(QQ,1)
            sage: H=Hom(P,P)
            sage: f=H([x^2+y^2,y^2])
            sage: f.is_morphism()
            True

        ::

            sage: P.<x,y,z>=ProjectiveSpace(RR,2)
            sage: H=Hom(P,P)
            sage: f=H([x*z-y*z,x^2-y^2,z^2])
            sage: f.is_morphism()
            False

        ::

            sage: R.<t>=PolynomialRing(GF(5))
            sage: P.<x,y,z>=ProjectiveSpace(R,2)
            sage: H=Hom(P,P)
            sage: f=H([x*z-t*y^2,x^2-y^2,t*z^2])
            sage: f.is_morphism()
            True
        """
        from sage.schemes.projective.projective_space import is_ProjectiveSpace
        if is_ProjectiveSpace(self.domain())==False or is_ProjectiveSpace(self.codomain())==False:
            raise NotImplementedError
        R=self.coordinate_ring()
        F=self._polys
        if R.base_ring().is_field():
            J=R.ideal(F)
        else:
            S=PolynomialRing(R.base_ring().fraction_field(),R.gens(),R.ngens())
            J=S.ideal([S.coerce(F[i]) for i in range(R.ngens())])
        if J.dimension()>0:
            return False
        else:
            return True

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def canonical_scheme(self, t=None):
        """
        Return the canonical scheme.

        This is a scheme that contains this hypergeometric motive in its cohomology.

        EXAMPLES::

            sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
            sage: H = Hyp(cyclotomic=([3],[4]))
            sage: H.gamma_list()
            [-1, 2, 3, -4]
            sage: H.canonical_scheme()
            Spectrum of Quotient of Multivariate Polynomial Ring
            in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring
            in t over Rational Field by the ideal
            (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^2*X1^3 + 27/64*Y0*Y1^4)

            sage: H = Hyp(gamma_list=[-2, 3, 4, -5])
            sage: H.canonical_scheme()
            Spectrum of Quotient of Multivariate Polynomial Ring
            in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring
            in t over Rational Field by the ideal
            (X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^3*X1^4 + 1728/3125*Y0^2*Y1^5)

        REFERENCES:

        [Kat1991]_, section 5.4
        """
        if t is None:
            t = FractionField(QQ['t']).gen()
        basering = t.parent()
        gamma_pos = [u for u in self.gamma_list() if u > 0]
        gamma_neg = [u for u in self.gamma_list() if u < 0]
        N_pos = len(gamma_pos)
        N_neg = len(gamma_neg)
        varX = ['X{}'.format(i) for i in range(N_pos)]
        varY = ['Y{}'.format(i) for i in range(N_neg)]
        ring = PolynomialRing(basering, varX + varY)
        gens = ring.gens()
        X = gens[:N_pos]
        Y = gens[N_pos:]
        eq0 = ring.sum(X) - 1
        eq1 = ring.sum(Y) - 1
        eq2_pos = ring.prod(X[i] ** gamma_pos[i] for i in range(N_pos))
        eq2_neg = ring.prod(Y[j] ** -gamma_neg[j] for j in range(N_neg))

        ideal = ring.ideal([eq0, eq1, self.M_value() * eq2_neg - t * eq2_pos])
        return Spec(ring.quotient(ideal))

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def __init__ ( self, base_ring_generators, generators,
                   relations, grading, all_relations = True, reduce_before_evaluating = True) :
        """
        The degree one part of the monomials that correspond to generators over the
        base expansion ring will serve as the coordinates of the elements.
        
        INPUT:
            - ``base_ring_generators``      -- A list of (equivariant) monoid power series with
                                               coefficient domain the base ring of the coefficient
                                               domain of the generators or ``None``.
            - ``generators``                -- A list of (equivariant) monoid power series; The generators
                                               of the ambient over the ring generated by the base ring
                                               generators.
            - ``relations``                 -- An ideal in a polynomial ring with ``len(base_ring_generators) + len(generators)``
                                               variables.
            - ``grading``                   -- A grading deriving from :class:~`fourier_expansion_framework.gradedexpansions.gradedexpansion_grading`;
                                               A grading for the polynomial ring of the relations.
            - ``all_relations``             -- A boolean (default: ``True``); If ``True`` the relations given
                                               for the polynomial ring are all relations that the Fourier
                                               expansion have.
            - ``reduce_before_evaluating``  -- A boolean (default: ``True``); If ``True`` any monomial
                                               will be Groebner reduced before the Fourier expansion
                                               is calculated.

        NOTE:
            The grading must respect the relations of the generators.
        
        TESTS::
            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_basicmonoids import *
            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_ring import *
            sage: from psage.modform.fourier_expansion_framework.monoidpowerseries.monoidpowerseries_element import *
            sage: from psage.modform.fourier_expansion_framework.gradedexpansions.gradedexpansion_grading import DegreeGrading
            sage: from psage.modform.fourier_expansion_framework.gradedexpansions.gradedexpansion_ring import *
            sage: mps = MonoidPowerSeriesRing(QQ, NNMonoid(False))
            sage: ger = GradedExpansionRing_class(Sequence([MonoidPowerSeries(mps, {1: 1}, mps.monoid().filter(4))]), Sequence([MonoidPowerSeries(mps, {1: 1, 2: 3}, mps.monoid().filter(4))]), PolynomialRing(QQ, ['a', 'b']).ideal(0), DegreeGrading((1,2)))
            sage: ger.base_ring()
            Graded expansion ring with generators a
       """
        if not hasattr(self, '_element_class') :
            self._element_class = GradedExpansion_class
        
        if hasattr(self, "_extended_base_ring") :
            Algebra.__init__(self, self._extended_base_ring)
        elif base_ring_generators is None or len(base_ring_generators) == 0 :
            Algebra.__init__(self, relations.base_ring())
        else :
            gb = filter( lambda p: all( all(a == 0 for a in list(e)[len(base_ring_generators):])
                                        for e in p.exponents() ),
                         relations.groebner_basis() )
            P = PolynomialRing( relations.base_ring(),
                                list(relations.ring().variable_names())[:len(base_ring_generators)] )
            base_relations = P.ideal(gb)
            R = GradedExpansionRing_class(None, base_ring_generators, base_relations,
                    grading.subgrading(xrange(len(base_ring_generators))), all_relations, reduce_before_evaluating)
            Algebra.__init__(self, R)

        GradedExpansionAmbient_abstract.__init__(self, base_ring_generators, generators, relations, grading, all_relations, reduce_before_evaluating)
        
        self._populate_coercion_lists_(
          coerce_list = [GradedExpansionBaseringInjection(self.base_ring(), self)],
          convert_list = [self.relations().ring()],
          convert_method_name = "_graded_expansion_submodule_to_graded_ambient_" )

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def get_ideal(self, name, base_ring=QQ, term_order="degrevlex"):
        """
        Returns the ideal given by 'name' over the base ring given by
        'base_ring' in a polynomial ring with the term order given by
        'term_order'.

        INPUT:

        - ``name`` - name as on the symbolic data website
        - ``base_ring`` - base ring for the polynomial ring (default: ``QQ``)
        - ``term_order`` - term order for the polynomial ring (default: ``degrevlex``)

        OUTPUT:

            ideal as given by ``name`` in ``PolynomialRing(base_ring,vars,term_order)``

        EXAMPLES::

            sage: sd = SymbolicData() # optional - database_symbolic_data
            sage: sd.get_ideal('Katsura_3',GF(127),'degrevlex') # optional - database_symbolic_data
            Ideal (u0 + 2*u1 + 2*u2 + 2*u3 - 1,
                   u1^2 + 2*u0*u2 + 2*u1*u3 - u2,
                   2*u0*u1 + 2*u1*u2 + 2*u2*u3 - u1,
                   u0^2 + 2*u1^2 + 2*u2^2 + 2*u3^2 - u0) of Multivariate Polynomial Ring in u0, u1, u2, u3 over Finite Field of size 127
        """

        def _getTextFromNode(node):
            t = ""
            for n in node.childNodes:
                if n.nodeType == n.TEXT_NODE:
                    t += str(n.nodeValue)
                else:
                    raise TypeError('not a text node')
            return t

        def _dom2ideal(node):
            """
            """
            l = []

            if str(node.nodeName) in ['vars','poly']:
                l.append(_getTextFromNode(node))

            for c in node.childNodes:
                l += _dom2ideal(c)

            return l

        orig_name = name
        name = name.replace('__','.')

        try:
            name = self.__intpath + name + ".xml"
            open(name)
        except IOError:
            try:
                name = self.__genpath + name + ".xml"
                open(name)
            except IOError:
                raise AttributeError("No ideal matching '%s' found in database."%orig_name)

        dom = parse(name)
        res = _dom2ideal(dom)
        variables, polys = res[0].replace("_",""), [p.replace("_","") for p in res[1:]]

        P = PolynomialRing(base_ring, len(variables.split(",")), variables)
        I = P.ideal([P(f) for f in polys])
        return I

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def points(self, B=0):
        r"""
        Return some or all rational points of an affine scheme.

        INPUT:

        - ``B`` -- integer (optional, default: 0). The bound for the
          height of the coordinates.

        OUTPUT:

        - If the base ring is a finite field: all points of the scheme,
          given by coordinate tuples.

        - If the base ring is `\QQ` or `\ZZ`: the subset of points whose
          coordinates have height ``B`` or less.

        EXAMPLES: The bug reported at #11526 is fixed::

            sage: A2 = AffineSpace(ZZ, 2)
            sage: F = GF(3)
            sage: A2(F).points()
            [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]

            sage: R = ZZ
            sage: A.<x,y> = R[]
            sage: I = A.ideal(x^2-y^2-1)
            sage: V = AffineSpace(R, 2)
            sage: X = V.subscheme(I)
            sage: M = X(R)
            sage: M.points(1)
            [(-1, 0), (1, 0)]

        ::

            sage: u = QQ['u'].0
            sage: K.<v> = NumberField(u^2 + 3)
            sage: A.<x,y> = AffineSpace(K, 2)
            sage: len(A(K).points(9))
            361

        ::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: E = A.subscheme([x^2 + y^2 - 1, y^2 - x^3 + x^2 + x - 1])
            sage: E(A.base_ring()).points()
            [(-1, 0), (0, -1), (0, 1), (1, 0)]
        """
        X = self.codomain()

        from sage.schemes.affine.affine_space import is_AffineSpace
        if not is_AffineSpace(X) and X.base_ring() in Fields():
            # Then X must be a subscheme
            dim_ideal = X.defining_ideal().dimension()
            if dim_ideal < 0: # no points
                return []
            if dim_ideal == 0: # if X zero-dimensional
                N = len(X.ambient_space().gens())
                S = X.defining_polynomials()[0].parent()
                R = PolynomialRing(S.base_ring(), 's', N, order='lex')
                phi = S.hom(R.gens(),R)
                J = R.ideal([phi(t) for t in X.defining_polynomials()])
                D = J.variety()
                points = []
                for d in D:
                    P = [d[t] for t in R.gens()]
                    points.append(X(P))
                points.sort()
                return points
        R = self.value_ring()
        if is_RationalField(R) or R == ZZ:
            if not B > 0:
                raise TypeError("a positive bound B (= %s) must be specified"%B)
            from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
            return enum_affine_rational_field(self,B)
        if R in NumberFields():
            if not B > 0:
                raise TypeError("a positive bound B (= %s) must be specified"%B)
            from sage.schemes.affine.affine_rational_point import enum_affine_number_field
            return enum_affine_number_field(self,B)
        elif is_FiniteField(R):
            from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
            return enum_affine_finite_field(self)
        else:
            raise TypeError("unable to enumerate points over %s"%R)

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def dual(self):
        r"""
        Return the projective dual of the given subscheme of projective space.

        INPUT:

        - ``X`` -- A subscheme of projective space. At present, ``X`` is
          required to be an irreducible and reduced hypersurface defined
          over `\QQ` or a finite field.

        OUTPUT:

        - The dual of ``X`` as a subscheme of the dual projective space.

        EXAMPLES:

        The dual of a smooth conic in the plane is also a smooth conic::

            sage: R.<x, y, z> = QQ[]
            sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
            sage: I = R.ideal(x^2 + y^2 + z^2)
            sage: X = P.subscheme(I)
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
              y0^2 + y1^2 + y2^2

        The dual of the twisted cubic curve in projective 3-space is a singular
        quartic surface. In the following example, we compute the dual of this
        surface, which by double duality is equal to the twisted cubic itself.
        The output is the twisted cubic as an intersection of three quadrics::

            sage: R.<x, y, z, w> = QQ[]
            sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
            sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
            sage: X = P.subscheme(I)
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 3 over
            Rational Field defined by:
              y2^2 - y1*y3,
              y1*y2 - y0*y3,
              y1^2 - y0*y2

        The singular locus of the quartic surface in the last example
        is itself supported on a twisted cubic::

            sage: X.Jacobian().radical()
            Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
            Polynomial Ring in x, y, z, w over Rational Field

        An example over a finite field::

            sage: R = PolynomialRing(GF(61), 'a,b,c')
            sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
            sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
            sage: X.dual()
            Closed subscheme of Projective Space of dimension 2 over
            Finite Field of size 61 defined by:
            y0^2 - 30*y1^2 - 20*y2^2

        TESTS::

            sage: R = PolynomialRing(Qp(3), 'a,b,c')
            sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
            sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
            sage: X.dual()
            Traceback (most recent call last):
            ...
            NotImplementedError: base ring must be QQ or a finite field
        """
        from sage.libs.singular.function_factory import ff

        K = self.base_ring()
        if not(is_RationalField(K) or is_FiniteField(K)):
            raise NotImplementedError("base ring must be QQ or a finite field")
        I = self.defining_ideal()
        m = I.ngens()
        n = I.ring().ngens() - 1
        if (m != 1 or (n < 1) or I.is_zero()
            or I.is_trivial() or not I.is_prime()):
            raise NotImplementedError("At the present, the method is only"
                                      " implemented for irreducible and"
                                      " reduced hypersurfaces and the given"
                                      " list of generators for the ideal must"
                                      " have exactly one element.")
        R = PolynomialRing(K, 'x', n + 1)
        from sage.schemes.projective.projective_space import ProjectiveSpace
        Pd = ProjectiveSpace(n, K, 'y')
        Rd = Pd.coordinate_ring()
        x = R.variable_names()
        y = Rd.variable_names()
        S = PolynomialRing(K, x + y + ('t',))
        if S.has_coerce_map_from(I.ring()):
            T = PolynomialRing(K, 'w', n + 1)
            I_S = (I.change_ring(T)).change_ring(S)
        else:
            I_S = I.change_ring(S)
        f_S = I_S.gens()[0]
        z = S.gens()
        J = I_S
        for i in range(n + 1):
#.........这里部分代码省略.........

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def _forward_image(self, f, check = True):
        """
        Compute the forward image of this subscheme by the morphism ``f``.

        The forward image is computed through elimination and ``f`` must be
        a morphism for this to be well defined.
        In particular, let $X = V(h_1,\ldots, h_t)$ and define the ideal
        $I = (h_1,\ldots,h_t,y_0-f_0(\bar{x}), \ldots, y_n-f_n(\bar{x}))$.
        Then the elimination ideal $I_{n+1} = I \cap K[y_0,\ldots,y_n]$ is a homogeneous
        ideal and $self(X) = V(I_{n+1})$.

        INPUT:

        - ``f`` -- a map whose domain contains ``self``

        - ``check`` -- Boolean, if `False` no input checking is done

        OUTPUT:

         - a subscheme in the codomain of ``f``.

        EXAMPLES::

            sage: PS.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: H = End(PS)
            sage: f = H([x^2, y^2-2*z^2, z^2])
            sage: X = PS.subscheme(y-2*z)
            sage: X._forward_image(f)
            Closed subscheme of Projective Space of dimension 2 over Rational Field
            defined by:
              y - 2*z

        ::

            sage: set_verbose(None)
            sage: PS.<x,y,z,w> = ProjectiveSpace(ZZ, 3)
            sage: H = End(PS)
            sage: f = H([y^2, x^2, w^2, z^2])
            sage: X = PS.subscheme([z^2+y*w, x-w])
            sage: f(X)
            Closed subscheme of Projective Space of dimension 3 over Integer Ring
            defined by:
              y - z,
              x*z - w^2

        ::

            sage: PS.<x,y,z,w> = ProjectiveSpace(CC, 3)
            sage: H = End(PS)
            sage: f = H([x^2 + y^2, y^2, z^2-y^2, w^2])
            sage: X = PS.subscheme([z-2*w])
            sage: f(X)
            Closed subscheme of Projective Space of dimension 3 over Complex Field
            with 53 bits of precision defined by:
              y + z + (-4.00000000000000)*w

        ::

            sage: R.<t> = PolynomialRing(QQ)
            sage: P.<x,y,z> = ProjectiveSpace(FractionField(R), 2)
            sage: H = End(P)
            sage: f = H([x^2 + 2*y*z, t^2*y^2, z^2])
            sage: f([t^2*y-z])
            Closed subscheme of Projective Space of dimension 2 over Fraction Field
            of Univariate Polynomial Ring in t over Rational Field defined by:
              y + (-1/t^2)*z

        ::

            sage: set_verbose(-1)
            sage: PS.<x,y,z> = ProjectiveSpace(Qp(3), 2)
            sage: H = End(PS)
            sage: f = H([x^2,2*y^2,z^2])
            sage: X = PS.subscheme([2*x-y,z])
            sage: f(X)
            Closed subscheme of Projective Space of dimension 2 over 3-adic Field
            with capped relative precision 20 defined by:
              z,
              x + (1 + 3^2 + 3^4 + 3^6 + 3^8 + 3^10 + 3^12 + 3^14 + 3^16 + 3^18 +
            O(3^20))*y

        ::

            sage: R.<y0,y1,y2,y3> = PolynomialRing(QQ)
            sage: P.<x,y,z> = ProjectiveSpace(FractionField(R), 2)
            sage: H = End(P)
            sage: f = H([y0*x^2+y1*z^2, y2*y^2+y3*z^2, z^2])
            sage: X = P.subscheme(x*z)
            sage: X._forward_image(f)
            Closed subscheme of Projective Space of dimension 2 over Fraction Field
            of Multivariate Polynomial Ring in y0, y1, y2, y3 over Rational Field
            defined by:
              x*z + (-y1)*z^2

            ::

            sage: P2.<x,y,z> = ProjectiveSpace(QQ, 2)
            sage: P5.<z0,z1,z2,z3,z4,z5> = ProjectiveSpace(QQ, 5)
            sage: H = Hom(P2, P5)
            sage: f = H([x^2,x*y,x*z,y^2,y*z,z^2]) #Veronese map
#.........这里部分代码省略.........

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def Chow_form(self):
        r"""
        Returns the Chow form associated to this subscheme.

        For a `k`-dimensional subvariety of `\mathbb{P}^N` of degree `D`.
        The `(N-k-1)`-dimensional projective linear subspaces of `\mathbb{P}^N`
        meeting `X` form a hypersurface in the Grassmannian `G(N-k-1,N)`.
        The homogeneous form of degree `D` defining this hypersurface in Plucker
        coordinates is called the Chow form of `X`.

        The base ring needs to be a number field, finite field, or `\QQbar`.

        ALGORITHM:

        For a `k`-dimension subscheme `X` consider the `k+1` linear forms
        `l_i = u_{i0}x_0 + \cdots + u_{in}x_n`. Let `J` be the ideal in the
        polynomial ring `K[x_i,u_{ij}]` defined by the equations of `X` and the `l_i`.
        Let `J'` be the saturation of `J` with respect to the irrelevant ideal of
        the ambient projective space of `X`. The elimination ideal `I = J' \cap K[u_{ij}]`
        is a principal ideal, let `R` be its generator. The Chow form is obtained by
        writing `R` as a polynomial in Plucker coordinates (i.e. bracket polynomials).
        [DalbecSturmfels]_.

        OUTPUT: a homogeneous polynomial.

        REFERENCES:

        .. [DalbecSturmfels] J. Dalbec and B. Sturmfels. Invariant methods in discrete and computational geometry,
           chapter Introduction to Chow forms, pages 37-58. Springer Netherlands, 1994.

        EXAMPLES::

            sage: P.<x0,x1,x2,x3> = ProjectiveSpace(GF(17), 3)
            sage: X = P.subscheme([x3+x1,x2-x0,x2-x3])
            sage: X.Chow_form()
            t0 - t1 + t2 + t3

        ::

            sage: P.<x0,x1,x2,x3> = ProjectiveSpace(QQ,3)
            sage: X = P.subscheme([x3^2 -101*x1^2 - 3*x2*x0])
            sage: X.Chow_form()
            t0^2 - 101*t2^2 - 3*t1*t3

        ::

            sage: P.<x0,x1,x2,x3>=ProjectiveSpace(QQ,3)
            sage: X = P.subscheme([x0*x2-x1^2, x0*x3-x1*x2, x1*x3-x2^2])
            sage: Ch = X.Chow_form(); Ch
            t2^3 + 2*t2^2*t3 + t2*t3^2 - 3*t1*t2*t4 - t1*t3*t4 + t0*t4^2 + t1^2*t5
            sage: Y = P.subscheme_from_Chow_form(Ch, 1); Y
            Closed subscheme of Projective Space of dimension 3 over Rational Field
            defined by:
              x2^2*x3 - x1*x3^2,
              -x2^3 + x0*x3^2,
              -x2^2*x3 + x1*x3^2,
              x1*x2*x3 - x0*x3^2,
              3*x1*x2^2 - 3*x0*x2*x3,
              -2*x1^2*x3 + 2*x0*x2*x3,
              -3*x1^2*x2 + 3*x0*x1*x3,
              x1^3 - x0^2*x3,
              x2^3 - x1*x2*x3,
              -3*x1*x2^2 + 2*x1^2*x3 + x0*x2*x3,
              2*x0*x2^2 - 2*x0*x1*x3,
              3*x1^2*x2 - 2*x0*x2^2 - x0*x1*x3,
              -x0*x1*x2 + x0^2*x3,
              -x0*x1^2 + x0^2*x2,
              -x1^3 + x0*x1*x2,
              x0*x1^2 - x0^2*x2
            sage: I = Y.defining_ideal()
            sage: I.saturation(I.ring().ideal(list(I.ring().gens())))[0]
            Ideal (x2^2 - x1*x3, x1*x2 - x0*x3, x1^2 - x0*x2) of Multivariate
            Polynomial Ring in x0, x1, x2, x3 over Rational Field
        """
        I = self.defining_ideal()
        P = self.ambient_space()
        R = P.coordinate_ring()
        N = P.dimension()+1
        d = self.dimension()
        #create the ring for the generic linear hyperplanes
        # u0x0 + u1x1 + ...
        SS = PolynomialRing(R.base_ring(), 'u', N*(d+1), order='lex')
        vars = SS.variable_names() + R.variable_names()
        S = PolynomialRing(R.base_ring(), vars, order='lex')
        n = S.ngens()
        newcoords = [S.gen(n-N+t) for t in range(N)]
        #map the generators of the subscheme into the ring with the hyperplane variables
        phi = R.hom(newcoords,S)
        phi(self.defining_polynomials()[0])
        #create the dim(X)+1 linear hyperplanes
        l = []
        for i in range(d+1):
            t = 0
            for j in range(N):
                t += S.gen(N*i + j)*newcoords[j]
            l.append(t)
        #intersect the hyperplanes with X
        J = phi(I) + S.ideal(l)
        #saturate the ideal with respect to the irrelevant ideal
        J2 = J.saturation(S.ideal([phi(t) for t in R.gens()]))[0]
#.........这里部分代码省略.........

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
    def points(self, B=0, prec=53):
        """
        Return some or all rational points of a projective scheme.

        INPUT:

        - ``B`` - integer (optional, default=0). The bound for the
          coordinates.

        - ``prec`` - he precision to use to compute the elements of bounded height for number fields.

        OUTPUT:

        A list of points. Over a finite field, all points are
        returned. Over an infinite field, all points satisfying the
        bound are returned.

        .. WARNING::

           In the current implementation, the output of the [Doyle-Krumm] algorithm
           cannot be guaranteed to be correct due to the necessity of floating point
           computations. In some cases, the default 53-bit precision is
           considerably lower than would be required for the algorithm to
           generate correct output.

        EXAMPLES::

            sage: P.<x,y> = ProjectiveSpace(QQ,1)
            sage: P(QQ).points(4)
            [(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
            (-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
            (1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
            (4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]

        ::

            sage: u = QQ['u'].0
            sage: K.<v> = NumberField(u^2 + 3)
            sage: P.<x,y,z> = ProjectiveSpace(K,2)
            sage: len(P(K).points(1.8))
            381

        ::

            sage: P1 = ProjectiveSpace(GF(2),1)
            sage: F.<a> = GF(4,'a')
            sage: P1(F).points()
            [(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]

        ::

            sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
            sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
            sage: E(P.base_ring()).points()
            [(-1 : -1 : 1), (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (0 : 1 : 1),
            (1 : -1 : 1), (1 : 0 : 1), (1 : 1 : 1)]
        """
        X = self.codomain()
        from sage.schemes.projective.projective_space import is_ProjectiveSpace
        if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
            #Then it must be a subscheme
            dim_ideal = X.defining_ideal().dimension()
            if dim_ideal < 1: # no points
                return []
            if dim_ideal == 1: # if X zero-dimensional
                rat_points = set()
                PS = X.ambient_space()
                N = PS.dimension_relative()
                BR = X.base_ring()
                #need a lexicographic ordering for elimination
                R = PolynomialRing(BR, N + 1, PS.gens(), order='lex')
                I = R.ideal(X.defining_polynomials())
                I0 = R.ideal(0)
                #Determine the points through elimination
                #This is much faster than using the I.variety() function on each affine chart.
                for k in range(N + 1):
                    #create the elimination ideal for the kth affine patch
                    G = I.substitute({R.gen(k):1}).groebner_basis()
                    if G != [1]:
                        P = {}
                        #keep track that we know the kth coordinate is 1
                        P.update({R.gen(k):1})
                        points = [P]
                        #work backwards from solving each equation for the possible
                        #values of the next coordinate
                        for i in range(len(G) - 1, -1, -1):
                            new_points = []
                            good = 0
                            for P in points:
                                #substitute in our dictionary entry that has the values
                                #of coordinates known so far. This results in a single
                                #variable polynomial (by elimination)
                                L = G[i].substitute(P)
                                if L != 0:
                                    L = L.factor()
                                    #the linear factors give the possible rational values of
                                    #this coordinate
                                    for pol, pow in L:
                                        if pol.degree() == 1 and len(pol.variables()) == 1:
                                            good = 1
#.........这里部分代码省略.........

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import ideal [as 别名]
            def expand(self, n, letter='x'):
                r"""
                Expand ``self`` written in the `\mathbf{w}` basis in `n^2`
                commuting variables which satisfy the relation
                `x_{ij} x_{ik} = 0` for all `i`, `j`, and `k`.

                The expansion of an element of the `\mathbf{w}` basis is
                given by equations (26) and (55) in [HNT06]_.

                INPUT:

                - ``n`` -- an integer
                - ``letter`` -- (default: ``'x'``) a string

                OUTPUT:

                - The symmetric function of ``self`` expressed in the ``n*n``
                  non-commuting variables described by ``letter``.

                REFERENCES:

                .. [HNT06] \F. Hivert, J.-C. Novelli, J.-Y. Thibon.
                   *Commutative combinatorial Hopf algebras*. (2006).
                   :arxiv:`0605262v1`.

                EXAMPLES::

                    sage: w = SymmetricFunctionsNonCommutingVariables(QQ).dual().w()
                    sage: w[[1,3],[2]].expand(4)
                    x02*x11*x20 + x03*x11*x30 + x03*x22*x30 + x13*x22*x31

                One can use a different set of variable by using the
                optional argument ``letter``::

                    sage: w[[1,3],[2]].expand(3, letter='y')
                    y02*y11*y20
                """
                from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
                from sage.combinat.permutation import Permutations

                m = self.parent()
                names = ['{}{}{}'.format(letter, i, j) for i in range(n) for j in range(n)]
                R = PolynomialRing(m.base_ring(), n*n, names)
                x = [[R.gens()[i*n+j] for j in range(n)] for i in range(n)]
                I = R.ideal([x[i][j]*x[i][k] for j in range(n) for k in range(n) for i in range(n)])
                Q = R.quotient(I, names)
                x = [[Q.gens()[i*n+j] for j in range(n)] for i in range(n)]
                P = SetPartitions()

                def on_basis(A):
                    k = A.size()
                    ret = R.zero()
                    if n < k:
                        return ret

                    for p in Permutations(k):
                        if P(p.to_cycles()) == A:
                            # -1 for indexing
                            ret += R.sum(prod(x[I[i]][I[p[i]-1]] for i in range(k))
                                         for I in Subsets(range(n), k))
                    return ret

                return m._apply_module_morphism(self, on_basis, codomain=R)