Python ProjectiveSpace.hom方法代码示例

# 需要导入模块: from sage.schemes.projective.projective_space import ProjectiveSpace [as 别名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import hom [as 别名]
    def parametrization(self, point=None, morphism=True):
        r"""
        Return a parametrization `f` of ``self`` together with the
        inverse of `f`.

        If ``point`` is specified, then that point is used
        for the parametrization. Otherwise, use ``self.rational_point()``
        to find a point.

        If ``morphism`` is True, then `f` is returned in the form
        of a Scheme morphism. Otherwise, it is a tuple of polynomials
        that gives the parametrization.

        ALGORITHM:

        Uses the PARI/GP function ``qfparam``.

        EXAMPLES ::

            sage: c = Conic([1,1,-1])
            sage: c.parametrization()
            (Scheme morphism:
              From: Projective Space of dimension 1 over Rational Field
              To:   Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
              Defn: Defined on coordinates by sending (x : y) to
                    (2*x*y : x^2 - y^2 : x^2 + y^2),
             Scheme morphism:
              From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
              To:   Projective Space of dimension 1 over Rational Field
              Defn: Defined on coordinates by sending (x : y : z) to
                    (1/2*x : -1/2*y + 1/2*z))

        An example with ``morphism = False`` ::

            sage: R.<x,y,z> = QQ[]
            sage: C = Curve(7*x^2 + 2*y*z + z^2)
            sage: (p, i) = C.parametrization(morphism = False); (p, i)
            ([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
            sage: C.defining_polynomial()(p)
            0
            sage: i[0](p) / i[1](p)
            x/y

        A ``ValueError`` is raised if ``self`` has no rational point ::

            sage: C = Conic(x^2 + 2*y^2 + z^2)
            sage: C.parametrization()
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

        A ``ValueError`` is raised if ``self`` is not smooth ::

            sage: C = Conic(x^2 + y^2)
            sage: C.parametrization()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
        """
        if (not self._parametrization is None) and not point:
            par = self._parametrization
        else:
            if not self.is_smooth():
                raise ValueError("The conic self (=%s) is not smooth, hence does not have a parametrization." % self)
            if point is None:
                point = self.rational_point()
            point = Sequence(point)
            Q = PolynomialRing(QQ, 'x,y')
            [x, y] = Q.gens()
            gens = self.ambient_space().gens()
            M = self.symmetric_matrix()
            M *= lcm([ t.denominator() for t in M.list() ])
            par1 = qfparam(M, point)
            B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
            # self is in the image of B and does not lie on a line,
            # hence B is invertible
            A = B.inverse()
            par2 = [sum([A[i,j]*gens[j] for j in range(3)]) for i in [1,0]]
            par = ([Q(pol(x/y)*y**2) for pol in par1], par2)
            if self._parametrization is None:
                self._parametrization = par
        if not morphism:
            return par
        P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
        return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)

# 需要导入模块: from sage.schemes.projective.projective_space import ProjectiveSpace [as 别名]
# 或者: from sage.schemes.projective.projective_space.ProjectiveSpace import hom [as 别名]
    def parametrization(self, point=None, morphism=True):
        r"""
        Return a parametrization `f` of ``self`` together with the
        inverse of `f`.

        If ``point`` is specified, then that point is used
        for the parametrization. Otherwise, use ``self.rational_point()``
        to find a point.

        If ``morphism`` is True, then `f` is returned in the form
        of a Scheme morphism. Otherwise, it is a tuple of polynomials
        that gives the parametrization.

        EXAMPLES:

        An example over a finite field ::

            sage: c = Conic(GF(2), [1,1,1,1,1,0])
            sage: c.parametrization()
            (Scheme morphism:
              From: Projective Space of dimension 1 over Finite Field of size 2
              To:   Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y
            + y^2 + x*z + y*z
              Defn: Defined on coordinates by sending (x : y) to
                    (x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
             Scheme morphism:
              From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y
            + y^2 + x*z + y*z
              To:   Projective Space of dimension 1 over Finite Field of size 2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (y : x))

        An example with ``morphism = False`` ::

            sage: R.<x,y,z> = QQ[]
            sage: C = Curve(7*x^2 + 2*y*z + z^2)
            sage: (p, i) = C.parametrization(morphism = False); (p, i)
            ([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
            sage: C.defining_polynomial()(p)
            0
            sage: i[0](p) / i[1](p)
            x/y

        A ``ValueError`` is raised if ``self`` has no rational point ::

            sage: C = Conic(x^2 + y^2 + 7*z^2)
            sage: C.parametrization()
            Traceback (most recent call last):
            ...
            ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + y^2 + 7*z^2 has no rational points over Rational Field!

        A ``ValueError`` is raised if ``self`` is not smooth ::

            sage: C = Conic(x^2 + y^2)
            sage: C.parametrization()
            Traceback (most recent call last):
            ...
            ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
        """
        if (not self._parametrization is None) and not point:
            par = self._parametrization
        else:
            if not self.is_smooth():
                raise ValueError("The conic self (=%s) is not smooth, hence does not have a parametrization." % self)
            if point is None:
                point = self.rational_point()
            point = Sequence(point)
            B = self.base_ring()
            Q = PolynomialRing(B, 'x,y')
            [x, y] = Q.gens()
            gens = self.ambient_space().gens()
            P = PolynomialRing(B, 4, ['X', 'Y', 'T0', 'T1'])
            [X, Y, T0, T1] = P.gens()
            c3 = [j for j in range(2,-1,-1) if point[j] != 0][0]
            c1 = [j for j in range(3) if j != c3][0]
            c2 = [j for j in range(3) if j != c3 and j != c1][0]
            L = [0,0,0]
            L[c1] = Y*T1*point[c1] + Y*T0
            L[c2] = Y*T1*point[c2] + X*T0
            L[c3] = Y*T1*point[c3]
            bezout = P(self.defining_polynomial()(L) / T0)
            t = [bezout([x,y,0,-1]),bezout([x,y,1,0])]
            par = (tuple([Q(p([x,y,t[0],t[1]])/y) for  p in L]),
                   tuple([gens[m]*point[c3]-gens[c3]*point[m]
                       for m in [c2,c1]]))
            if self._parametrization is None:
                self._parametrization = par
        if not morphism:
            return par
        P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
        return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)